REACTIVE FORCE FIELD BASED ATOMISTIC SIMULATIONS OF SILICON ANODE UPON LITHIATION AND DELITHIATION IN LITHIUM - ION BATTERIES By Kwang Jin Kim A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Materials Science and Engineering Doctor of Philosophy 201 8 ABSTRACT REACTIVE FORCE FIELD BASED ATOMISTIC SIMULATIONS OF SILICON ANODE UPON LITHIATION A ND DELITHIATION IN LITHIUM - ION BATTERIES By Kwang Jin Kim Silicon (Si) has been considered as a promising anode material for lithium - ion batteries due to its high theoretical capacity (3750 mAh/g), low discharge voltage, abundancy, and low cost. However, e lectrochemical lithiation and delithiation of Si proceeds via solid - state amorphization and massive volume expansion/contraction, resulting in destructive consequences such as slow rate performance, irreversible capacity loss, and mechanical degradation. T hese problems significantly affect the capacity retention and cycle life and limits the wide application of Si anode. In this thesis, molecular dynamic (MD) simulations with reactive force field (ReaxFF) were performed to better understand and design optim ized Si anodes with enhanced rate performance and minimized irreversible capacity loss. Furthermore, the transferability of ReaxFF to simulate SiO system was evaluate and the ground work was laid to design extensive training set of Li - Si system for machine - learning potentials development. There are two major discoveries based on the simula tion work. First, t o elucidate the rate - limiting factor upon lithiation of Si for improved rate performance, reactive MD simulations were performed in c rystalline - Si and a morphous - Si at the atomic - scale. It was discovered that Si movement is the rate - limiting factor . It was also revealed that Li diffusivity increases with Li concentrations, opposite to many currently used intercalation compounds. Furthermore, the new findi ng highlight ed that vacancies in Si can accelerate the lithiation process dramatically. Then, t he irreversible atomic - scale structural changes upon delithiation was studied using a newly - developed reactive MD - based delithiation algorithm with well - controlled chemical potential gradient driving force and delithiation rate. D uring fast delithiation, a cage - like structure with high Si content was formed near the surface, thus trapping significant amount of Li atoms inside the Si - thin - film. Furthermore, delithiated amorphous Li x Si (with no porosity and trapped Li) still ha s high er volume (lower density) than the equilibrium structures at the same Li concentration throughout the whole delithiation process regardless of the delithiation rates. The origin of the excess volume is the loss of directly bonded Si - Si pairs, which makes t he subsequent re - lithiation proceed faster. These new insights lead to several recommendations, such as the delithiation rate and depth of charge, to avoid trapped Li and coating delamination in order to enhance the life of Si electrodes. iv ACKNOWLEDGEMENTS First of all, I would like to thank my advisor Prof. Yue Qi for her great inspirations and motivations. She has provided constructive guidance, which trained me to become an independent researcher. Without her constant encouragements, I would not be able to complete my research journey. Het integrity, enthusiasm, and companionship set an excellent example for me to follow, which will enlighten and guide my life. I also want to appreciate my committee member, Prof. Wei Lai, Prof. Lawrence Dzral, and Prof. Rebecca Anthony for their helpful directions and suggestions on my researches. Additionally, special thanks to our collaborator, Prof. H.Metin Aktulga and Prof. Matthew Hirn, for their helpful discuss ion and contributions. I also want to acknowledge all my colleagues and friends in Michigan State University: Christine James, Tridip Das, Yuxiao Lin, Jialin Liu, Hong - Kang Tian, Thanaphong Phongpreecha, Brumwell. They shared their expertise with me and support my research. It has been a pleasure to know you and share my life in graduate school. Most importantly, I want to appreciate my family for their unconditional love and encouragements. Their trust and love made me a strong and confident person, which allowed me to complete this work. v TABLE OF CONTENTS LIST OF TABLES ................................ ................................ ................................ ........................ vii LIST OF FIGURES ................................ ................................ ................................ ..................... viii Chapter 1 Background and Motivations ................................ ................................ ......................... 1 1.1 Li - ion Batteries and Silicon Negative Electrode ................................ ............................. 1 1.1.1 Thermodynamics and Kinetics of Electrochemical Reaction of Li with Si ............. 6 1.1.2 Computational Studies of Li - Si System ................................ ................................ . 11 1.1.2. (a) Density Functional Theory Calculation of Li - Si ................................ ............... 12 1.1.2. (b) Molecular Dynamics Simulation of Li - Si ................................ ......................... 16 1.2 Motivation and Thesis Outline ................................ ................................ ...................... 18 Chapter 2 Overview of Computational Methods ................................ ................................ .......... 20 2.1 Energy Calculation using DFT and Force Fields ................................ .......................... 20 2.1.1 Density Functional Theory ................................ ................................ ..................... 20 2.1.2 Force Fields ................................ ................................ ................................ ............ 22 2.1.2. (a) Introduction ................................ ................................ ................................ .. 22 2.1.2. (b) Classical Force Field ................................ ................................ .................... 23 2.1.2. (c) Reactive Force Field ................................ ................................ ..................... 25 2.2 Molecular Dynamics ................................ ................................ ................................ ..... 26 2.2.1 Ab Initio MD and Force Field based MD ................................ ............................... 26 2.2.2 Equation of Motions and Thermodynamic Ensembles ................................ ........... 28 Chapter 3 Vacancies in Si Can Improve the Co ncentration - Dependent Lithiation Rate .............. 30 3.1 Summary ................................ ................................ ................................ ....................... 30 3.2 Introduction ................................ ................................ ................................ ................... 31 3.3 Simulation Methods ................................ ................................ ................................ ...... 33 3.3.1 Reactive Force Field for Li - Si System ................................ ................................ ... 33 3.3.2 Molecular Dynamics Simulations ................................ ................................ .......... 34 3.3.3 Local Concentration and Diffusion Properties Analysis ................................ ........ 36 3.4 Results and Discussion ................................ ................................ ................................ . 38 3.4.1 How Lithiation Proceeds in a - Si ................................ ................................ ............. 39 3.4.2 How Lithiation Proceeds in c - Si ................................ ................................ ............. 39 3.4.3 Si Vacancy Generation Accelerates Lithiation Dynamics ................................ ..... 41 3.4.4 Random Diffusion of Li in Si is Concentration Dependent ................................ .. 46 3.4.4. (a) Local Concentration Evolution ................................ ................................ ..... 46 3.4.4. (b) Random Diffusion of Li in Si is Concentration - Dependent ......................... 49 3.4.4. (c) Room - Temperature Li and Si Diffusivity Calculation ................................ . 51 3.5 Conclusions ................................ ................................ ................................ ................... 52 Chapter 4 Atomistic Simulation Derived Insight on the Irreversible Structural Changes of Si Electrode during Fast and Slow Delithiation ................................ ................................ ................ 54 4.1 Summary ................................ ................................ ................................ ....................... 54 vi 4.2 Introduction ................................ ................................ ................................ ................... 55 4.3 Simulation Methods ................................ ................................ ................................ ..... 58 4.3.1 Reactive Force Field for Li - Si - Al - O - H System ................................ ..................... 58 4.3.2 Molecular Dynamics Simulations of the Delithiation Process ............................... 60 4.3.3 Analysis of Volu me Contraction, Pore Evolution, and Li Trapping ...................... 63 4.3.4 Comparison between a - Li x Si at Equilibrium and Relithiated a - Li x Si .................... 65 4.4 Results and Discussion ................................ ................................ ................................ .. 66 4.4.1 Delithiation Proceeds with Different Rates ................................ ............................ 66 4.4.2 Porous Structures Evolut ion and Coating Delamination ................................ ........ 68 4.4.3 Quantifying the Volume Contribution ................................ ................................ .... 71 4.4.4 Dilated a - Li x Si Exhibits Faster Lithiation Rate in the Second Cycle ..................... 73 4.4.5 Si Cage as the Origin of Li Trapping - Induced Irreversible Capacity Loss ............ 75 4.5 Conclusions ................................ ................................ ................................ ................... 77 Chapter 5 Reactive Force Field Evaluation for Li thiation/Delithiation in Si - O ........................... 79 5.1 Introduction ................................ ................................ ................................ ................... 79 5.2 Simulation Methods ................................ ................................ ................................ ...... 82 5.2.1 First - principles DFT Calculations ................................ ................................ .......... 82 5.2.2 ReaxFF - based Molecular Dynamics Calculations ................................ ................. 83 5.3 Results and Discussions ................................ ................................ ............................... 85 5.3.1 Direct Energy Comparison of ReaxFF versus DFT ................................ ............... 85 5.3.2 Non - Physical SiO Structure Predicted by ReaxFF - MD ................................ ......... 87 5.3.3 Reason for the Errors in the MD Results ................................ ............................... 89 5.3.4 Design of the Training Set for Li - Si - O System ................................ ...................... 90 5.4 Conclusions ................................ ................................ ................................ .................. 91 Chapter 6 Generation of Li - Si Training Set for Machine - Learning Potential .............................. 93 6.1 Introduction ................................ ................................ ................................ ................... 93 6.2 Automated Construction of the Training Set with Crystalline and Amorphous Li x Si .. 94 6.2.1 c - Li x Si structures ................................ ................................ ................................ .... 94 6.2.2 a - Li x Si Structures ................................ ................................ ................................ ... 95 6.2.3 Formation Energy and OCV for the Training Set ................................ .................. 99 6.3 Elastic Property Calculation to Validate Machine Learning Results .......................... 102 6.4 Conclusions ................................ ................................ ................................ ................. 104 Chapter 7 Conclusions ................................ ................................ ................................ ................ 106 APPEND IX ................................ ................................ ................................ ................................ . 109 REFERENCES ................................ ................................ ................................ ........................... 137 vii LIST OF TABLES Table 3 - 1 Summary of time required to reach fully lithiated stage, together with simulation information of amorphous Si and crystalline Si with (100), (110), and (111) surface orientations ................................ ................................ ................................ ................................ ....................... 35 Table 5 - 1 Summary of density, microstructure, and simulation size of a - Si, a - SiO, and a - SiO 2 84 Table 5 - 2 Summary of energy calculation of a - Si, a - SiO, and a - SiO 2 from optimized structures generated by first - principle DFT calculations and ReaxFF - MD simulations ............................... 86 Table 6 - 1 Structural parameters for c - Li, c - Si, and c - Li x Si alloys at equilibrium ....................... 95 viii LIST OF FIGURES Figure 1 - 1 Energy density comparison of different batteries systems (Adapted from reference [ 1 ]. Copyright © Nature 2001) ................................ ................................ ................................ .............. 1 Figure 1 - 2 Schematic diagram of Li - ion batteries, which is composed of cathode (LiCoO 2 ), anode (graphite), electrolyte, and separator (Adapted from reference [ 3 ]. Copyright © Journal of Power Sources 2013) ................................ ................................ ................................ ................................ .. 2 Figure 1 - 3 (a) Gravimetric capacity of different anode materials (Adapted from reference [ 6 ]. C opyright © Chemical Reviews 2014) (b) Degradation mechanism of Si anodes upon lithiation - driven volume expansion (Adapted from reference [ 7 ]. Copyright © Nature Reviews Materials 2016) ................................ ................................ ................................ ................................ ............... 3 Figure 1 - 4 (a) - (c) Various morphologies of Si active materials and their composites (Adapted from reference [ 7,12 ]. Copyright © Nature Reviews Materials 20 16) (d) Schematic of SiO x composite anode in atomic - level (Adapted from reference [ 13 ]. Copyright © The Journal of Physical Chemistry C 2016) (e) Schematic describing the formation of the artificial SEI on Li x Si nanosphere (Adapted from reference [ 14 ]. Cop yright © Journal of the American Chemical Society 2015) (f) Illustration of Si binder concepts (Adapted from reference [ 7 ]. Copyright © Nature Reviews Materials 2016) ................................ ................................ ................................ ................ 5 Figure 1 - 5 (a) Li - Si phase diagram (Adapted from reference [ 40 ]. Copyright © Bulletin of Alloy Phase Diagram 1990) (b) Experimentally measured voltage vs. composition of Li - Si system. .... 6 Figure 1 - 6 (a) Schematic diagram of phase evolution upon multiple cycle based on in situ XRD of charge/discharge cycles of Li - Si battery (Adapted from refere nce [ 33 ]. Copyright © Journal of the Electrochemical Society 2007) (b) High - resolution TEM image of the sharp phase boundary between c - Si and a - Li x Si (Adapted from reference [ 37 ]. Copyright © Physical Review Letters 2011) ................................ ................................ ................................ ................................ ......................... 8 Figure 1 - 7 (a) Characteristic volume expansion of c - Si nanopillars along <100>, <110>, and <111> axial orientations upon lithiation (Adapted from reference [ 45 ]. Copyright © Nano Letters 2011) (b) High - resolution TEM image and schematic diagram describing the lateral ledge flow along {111} plane (Adapted from reference [ 43 ]. Copyright © Nature Nanotechnology 2012) .... 9 Figure 1 - 8 Multi - scale modeling and simulation ................................ ................................ ........ 11 Figure 1 - 9 (a) Experimentally measured (red curve) and computationally calculated (blue, green, and black curve) voltage vs. composition of Li x Si. (Adapted from reference [ 52 ]. Copyright © Journal of the Electrochemical Society 2009) (b) Net charge of Li and Si in c - Li x Si (LiSi, Li 12 Si 7 , Li 7 Si 3 , Li 13 Si 4 , Li 15 Si 4 , and Li 21 Si 5 ). (Adapted from reference [ 54 ]. Copyright © Journal of Alloys and Compounds 2010) ................................ ................................ ................................ .................. 13 ix Figure 1 - 10 (a) Representative structural snapshots at various lithiation stages of c - Si (top) and a - Si (a - Si) at 1200K. Cyan and purple color spheres in the figure represent Si and Li atoms, respectively (b) Representation of structures form during lithiation (c) Structural statistics with time upon for lithiation of c - Si. (Adapted from reference [ 56 ]. Copyright © Nano Letters 2011) 15 Figure 1 - 11 (a) Structural snapshot representing the lithiation of a - Si and c - Si with (100), (110), and (111) surface orientations after 200 ps at 1200K usin g ReaxFF - MD simulations (Adapted from reference [ 62 ]. Copyright © The Journal of Physical Chemistry C 2014) (b) Structural snapshot representing the delithiation of lithiated Si NW with MD equilibration time of 250 ps (red) and 500 ps (black) for each deli thiation steps (Adapted from reference [ 63 ]. Copyright © The Journal of Physical Chemistry C 2015) ................................ ................................ ................................ ..... 16 Figure 2 - 1 Description of e nergy terms for classical force fields. Each number represents (1) Bond, (2) Angle, (3) Dihedral, (4) van der Waals, and (5) Electrostatic contribution ................. 24 Figure 2 - 2 (a) Overview of the ReaxFF total energy components 61 (b) Interatomic distance dependency of the carbon - carbon bond order 72 ................................ ................................ ............ 25 Figure 2 - 3 Illustration of molecular dynamics algorithm ................................ ........................... 28 Figure 3 - 1 Energy versus volume of (a) Various Li crystals and (b) crystalline LiSi and Li 13 Si 4 predicted by DFT and ReaxFF calculation (Adapted from referen ce [ 59 ]. Copyright © Modelling and Simulation in Materials Science and Engineering 2013) ................................ ....................... 34 Figure 3 - 2 (a) ~ (e) Structural snapshots and local concentrations profile at successive stages of lithiation of a - Si and (f) ~ (j) c - Si with (100) surface orientation. All snapshots are taken at 1200 K. In the figure, spheres in blue and green color represent Si and Li atoms, respective ly ........... 38 Figure 3 - 3 Structural snapshots of c - Si with (100) surface orientations with (a) no vacancy, (b) 5 % Li vacancy, (c) and 5 % Si vacancy after a reaction time of 10 ps at 1200 K. In the figure, spheres in blue and green color represent Si and Li atoms, respectively (d) The concentration profiles of the three snapshots. (e) Movement of reaction fronts with and without Si vacanci es with respect to time at 1200 K ................................ ................................ ................................ .............. 42 Figure 3 - 4 (a) Movement of reaction fronts with 2 %, 4 %, 6 %, 8 %, and 10 % Si vacancies w ith respect to time (b) Reaction rates with corresponding Si vacancy concentrations. All information was obtained during lithiation of c - Si with (100) orientation at corresponding Si vacancy concentration at 1500K ................................ ................................ ................................ ................. 43 Figure 3 - 5 Radial distribution functions g Si - Si (r) at 1500K for c - Si with (100) orientation with 2 % and 8 % Si vacancies at (a) 1 ps and (b) 1 ns ................................ ................................ ................ 45 Figure 3 - 6 Local concentration distributions with respect to (a) position at different time and (b) time at different positions at 1500 K. Concentration distributions respect to (c) position at different x time and (d) time at different positions at 1200 K. All conce ntration profile was obtained during lithiation of c - Si with (100) orientation at corresponding temperature ................................ ........ 47 Figure 3 - 7 (a) Di ffusion coefficients of Li at different concentrations in logarithmic scale at 1200 K. Diffusion coefficients with corresponding concentration upon lithiation from bulk a - Li x Si is represented by blue dots. Red dots represent data obtained from local MSD ca lculations from initially c - Si with (100) surface orientation (b) Comparison Li diffusivity at with first - principle calculation and experiment ................................ ................................ ................................ ........... 49 Figure 4 - 1 Model setup . (a) Initial structure of fully lithiated a - Li 3.75 Si thin film sandwiched between fully lithiated aluminum oxide a - Li 8 Al 2 O 3 coatings, where spheres in yellow, green, red, and blue represent Si, Li, O, and Al atoms re spectively (b) The computed open circuit voltage for a - Li x Si and a - Li x Al 2 O 3 . Continuously removing Li from the coating layer outside of the buffer zone (white region) will generate the chemical potential driving force for the Li inside Si thin film naturally ................................ ................................ ................................ .............. 60 Figure 4 - 2 Schematic representation of Connolly volume calculation with (a) probe (b) structure with in ternal pore smaller than single atom size and (c) system with internal pore larger than single atom size (considered as isolated - inner pore, region in yellow) ................................ ................... 65 Figure 4 - 3 Atomic Structure Evolution. Structural snapshots at the specified delithiation steps during the continuous delithiation process with the (a) - (c) fast and (d) - (f) slow rates. In the figure, spheres in yellow, green, red, and blue represent Si, Li, O, and Al atoms, respectively .. 66 Figure 4 - 4 Pore Structure Evolution. Structural snapshots of the pore structure and distribution in the highlighted a - Li x Si region during continuous delithiation process with the (a) fast and (b) slow rate ................................ ................................ ................................ ................................ ........ 68 Figure 4 - 5 Volume and Concentration Evolution. (a) The total volume (V total ) and truly occupied a - Li x Si volume (V true ) change normalized with respect to the unlithiated Si and (b) Li concentration and the pore volume contribution (V pore / V total ), upon delithiation with the fast and slow rate ................................ ................................ ................................ ................................ ........ 71 Figure 4 - 6 Difference in Amorphous Structures. (a) a - Li x Si structures formed during delithiation have lower density than the equilibrium structures. Density corresponding to the delithiated a - Li x Si upon delithiation with fast and slow rate are r epresented by blue and red dots. Green dots represent density obtained from the equilibrium a - Li x Si structures (b) RDF of g Si - Si (r) at 900 K of delithiated a - Li x Si upon delithiation with fast and slow rate and equilibrium a - Li x Si, all at concentration 2.8 . Local concentration distributions at different stages of lithiation of (c) delithiated a - Si with slow rate and (d) equilibrium a - Si at 1200 K ................................ .............. 73 Figure 4 - 7 Si Cage and Trapped Li. (a) Local Li concentration distribution at the final stage of delithiation with the fast (55 step) and slow (65 step) rate (b) Li motion tracking with various initial positions during fast delithiation. In the figure, Si and Li atoms are excluded to improve the clarity. Spheres in green, red, and blue represent Li, O, Al atoms, respectively ................................ ...... 75 xi Figure 5 - 1 (a) EELS profiles (Si - K edge) taken from the three different regions in the HAADF - STEM image (b) Experimental (a) - (c) and simulated (d) - (f) ABED images corresponding to Si ,SiO 2 , and interface regions (Adapted fr om reference [ 133 ]. Copyright © Nature Communications 2016) ................................ ................................ ................................ ................. 80 Figure 5 - 2 Schematic of (a) random bonding and (b) random mixture a - SiO ............................ 83 Figure 5 - 3 Energy calculation of a - SiO 0.5 , a - SiO, and a - SiO 2 wi th equation of states ............... 85 Figure 5 - 4 Formation energy of a - SiO with (a) DFT optimized and (b) ReaxFF - MD optimized structure. The red and blue data points represent DFT and ReaxFF - MD energy calculation, respectively ................................ ................................ ................................ ................................ ... 86 Figure 5 - 5 (a) Energy Comparison of ran dom bonding and random mixture a - SiO with its unique microstructure. (a) - # represents the microstructure of a - SiO corresponding to the energy. In the f ) of a - SiO structures ................................ ................................ ................................ ........................ 88 Figure 5 - 6 Energy calculation using DFT and ReaxFF methods for (a) three different Si phases and (b) various silicon dioxide phases at different densities fo r ReaxFF fitting (Adapted from reference [ 73 ]. Copyright © The Journal of Physical Chemistry A 2003) ................................ .... 89 Figure 6 - 1 Flow chart des cribing the automated protocol to construct the Li - Si training set .... 96 Figure 6 - 2 (a) Radial distribution function of c - Li 15 Si 4 and a - Li 15 Si 4 (b) Densities of a - Si, a - LiSi, a - Li 12 Si 7 , a - Li 7 Si 3 , a - Li 13 Si 4 , and a - Li 15 Si 4 calculated from ReaxFF - NPT simulations via melting - and - quenching process ................................ ................................ ................................ .................. 97 Figure 6 - 3 Formation energies of crystalline and amorphous Li x Si included in the training set. The training set is consisted of total 88201 structures ................................ ................................ 100 Figure 6 - 4 Experimental 34,42 and calculated OCV vs. composition curves of Li/Si system at high temperature (lines in red) and room temperature (curves in blue) (Adapted from reference [ 34 ]. Copyright © Journal of the Electrochemical Society 2004) ................................ ....................... 101 Figure 6 - 5 Bulk modulus for a - Li x Si with comparison with literatu re (DFT) 49,55 .................... 104 1 Chapter 1 Background and Motivations 1.1 Li - ion Batteries and Silicon Negative Electrode Figure 1 - 1 Energy density comparison of different batteries systems ( Adapted from reference [ 1 ]. Copyright © Nature 2001) In the past few decades, Lithium - ion batteries have become the primary energy storage devices and power sources for portable electronics, power tools, and hybrid/full electric vehicles due to its high energy density, as shown in Fig. 1 - 1 . To meet the incre asing demand for large - scale energy storage , particularly for the applications to increasingly popular electric vehicles, it is necessary to significantly improve Li - ion batteries in terms of energy density, specific capacity, durability, safety, and cost. 1 3 As represented in Fig. 1 - 2 , in commercial Li - ion batteries, transition metal oxides or phosphates (LiCoO 2 , LiMnO 2 , LiFePO 4 etc.) are commonly used as the cathode (positive electrode) materials, while graphite is commonly used as the anode (negative electrode) material. 2 Figure 1 - 2 Schematic diagram of Li - ion batteries, which is composed of cathode (LiCoO 2 ), anode (graphite), electrolyte, and separator (Adapted from reference [ 3 ]. Copyright © Journal of Power Sources 2013) Cathode and anode are separated by a porous separator filled with electrolytes, which prevents the electrical contact between the electrodes and allows Li - ion diffusio n during charging and discharging process. In electrochemistry, anode refers to the electrode where oxidation is taking place. However, in the rechargeable battery system, oxidation takes place in both electrodes depending on whether the system is under ch arging and discharging. Therefore, in this thesis, anode is designated to be the electrode where oxidation is taking place during the discharge. During the charging process, two electrodes are externally connected to an electrical supply and electrons are released and externally moves from the cathode to the anode. Simultaneously, Li ions internally move in the same direction through the electrolyte and store the external energy in the form of chemical energy. The energy released during b attery charging and discharging can be represented by the following equation (1 - 1) where V is the voltage and q is the amount of charge transferred. 4 Since the amount of charge that can be reversibly stored in the electrodes (number of Li or electrons released from electrochemical 3 reaction at the electrodes) determines the battery capacity, developing novel electrode materials is crucial to enhance the battery energy density . In terms of anode, graphite interacts with Li ions via an intercalation mechanism, where Li ions are insert ed into and extracted from the interstitial sites provided by the layered graphite. The number of limited intercalation sites prevents significant structural changes, resulting in good capacity retention upon cycling. However, the limited number of interca lation sites within the graphite confines the theoretical specific capacity to 372 mAh/g. 1 The need for alternative anode materials with improved specific capacity motivated researchers to investigate novel electrode materials which interact wit h Li via an alloying mechanism. In contrast to intercalation mechanisms, alloying mechanisms proceeds via breakage of the bonds between the host atoms and formation of Li alloys. Since the reaction of the host material and Li are not limited by the atomic framework of the host material, significantly more Li atoms can be stored compared to intercalation electrode materials. 5,6 Figure 1 - 3 (a) Gravimetric capacity of different anode materials (Adapted from reference [ 6 ]. Copyright © Chemical Reviews 2014) (b) Degrad ation mechanism of Si anodes upon lithiation - driven volume expansion (Adapted from reference [ 7 ]. Copyright © Nature Reviews Materials 2016) 4 Among many alloying anode materials, Silicon (Si) is considered as a promising candidate because of its exceptionally high theoretical specific capacity (3750 mAh/g). Despite its high specific capacity, Si is limited in its application because of its 300 % volume expansion and massive structural change during lithiation/delithiation, caused by the large number of Li reacting with the Si. 8 This leads to mechanical fracture, formation of unstable solid electrolyte interphase (SEI), and disconnection between the particles, which all t ogether contribute to the irreversible capacity loss and limited cycle life. 7,9 12 To facilitate the practical implementation of Si anodes, various strategies have been employed to mitigate the mechanical degradation and improve t he rate performance and capacity retention. 7,12,23,15 22 As shown in Fig. 1 - 4 (a) , designing effective nano - structured Si ( Si nanoparticles, 19 Si nanowires, 17, 18 Si nanotubes, 21,22 Si nanospheres 23 ) is a promising direction since they can efficiently alleviate the effect of dramatic volume expansion, which results in enhanced capacity retention and improved cycle life. However, reducing the size of Si to nano - domain induces another significant problem; increasing formation of Solid Electrolyte Interphase (SEI). 24 26 Due to high surface - to - volume ratio of nanostructures, much more surfaces are exposed to the electrolyte and form SEI layer. Furthermore, huge volume change of nanostructured Si continuously breaks the SEI layer and generate new surface exposed to the ele ctrolyte, thereby constantly consuming active Li which causes rapid capacity loss and low current efficiency. To solve challenges rising from reducing the size of Si to nano - domain, researchers further engineered the nano - structures (Fig . 1 - 4 ( c )) with int entionally designed voids 19 and dev eloped composite materials in some form of carbon 27 or TiO 2 28 (Fig . 1 - 4 (b)) . These unique designs accommodate the volume expansion without any outward expansion, thereby avoiding direct contact with the electrolyte and alleviating the SEI problem. 5 Figure 1 - 4 (a) - (c) Various morp hologies of Si active materials and their composites (Adapted from reference [ 7,12 ]. Copyright © Nature Reviews Materials 2016) (d) Schematic of SiO x composite anode in atomic - level (Adapted from reference [ 13 ]. Copyright © The Journal of Physical Chemistry C 2016) (e) S chematic describing the formation of the artificial SEI on Li x Si nanosphere (Adapted f rom reference [ 14 ]. Copyright © Journal of the American Chemical Society 2015) (f) Illustration of Si binder concepts (Adapted from reference [ 7 ]. Copyright © Nature Reviews Materials 2016) Utilizing composite form at atomic - level, such as Silicon Monoxide (SiO) 29 , is another strategy to mitigate the effect of volume expansion (Fig . 1 - 4 (d)) . SiO is composed of inhomogeneous mixture of a - Si an d a - SiO 2 in atomic - level where Li reacts with SiO to form several irreversible phases (Li 2 Si 2 O 5 , Li 6 Si 2 O 7 , Li 4 SiO 4 , and Li 2 O), which act as buffering phases against the volume expansion during cycling. Even though the formation of irreversible phase 6 results in lower discharge capacity (~ 2200 2500 mAh/g) compared to Si, significantly reduced volume expansion (~ 160 %) enhances the capacity retention and cycle life of SiO. Usage of electrolyte additives and generation of effective surface passivation layers (artificial SEI layer) 14 could be a valuable strategy to prevent the chemical degradation (Fig . 1 - 4 (e)) . Finally, utilizing conductive binders, which is stiff, inert to electrolyte, and conductive to Li ions and helpful to stabilize SEI, also dramatically improve the cycle life (Fig . 1 - 4 (f)) . 7,30 For detailed information, see the review article by Choi et al . and Li et al . and the references therein. 7,12 Along with the massive volume expansion, concurrently occurring upon electrochemical lithiation and d elithiation of Si is the crystalline - to - amorphous phase transition. 31 39 Solid - state amorphization results in characteristic reversible and irreversible structural evolution, chemical reactions, and diffusion of active materials which significantly affects the irreversible capacity loss, stress generation, and rate performance. Therefore, it is also important to investigate the thermodynamics and kinetics during electrochemical lithiation/delithiation of Si, which can be utiliz e d to design battery with improved per fo rmance. 1.1.1 Thermodynamics and Kinetics of Electrochemical Reaction of Li with Si Figure 1 - 5 (a) Li - Si phase diagram (Adapted from reference [ 40 ]. Copyright © Bulletin of Alloy Phase Diagram 1990 ) ( b ) Experimentally measured voltage vs. composition of Li - Si system. 7 Figure 1 - Dotted curve represent s the coulometric titration curve of Li x Si at 415 C and solid curve represents the galvanostatic charge / discharge profile of Si powder electrode at room temperature ( Adapted from reference [ 41,42 ]. Copyright © Journal of Solid State Chemistry 1981) The reaction of Li and Si follows the eq uilibrium phase diagram ( Fig. 1 - 5 (a) ) at high temperature, forming intermetallic compounds with nominal composition of Li 12 Si 7 , Li 7 Si 3 , Li 13 Si 4 , and Li 22 Si 5 . 40 The formation of equilibrium intermetallic compounds were confirmed by Wen and Huggins , 42 who - KCl(eut.)|Li y Si(s) to study the alloying process of Li - Si at high temperature (415 o C) using equilibrium coulo metric titration technique. They observed four distinct voltage plateaus ( Fig. 1 - 5 (b), dotted curve ), which clearly indicated the sequential formation of equilibrium intermetallic compounds corresponding to the composition of Li 12 Si 7 , Li 7 Si 3 , Li 13 Si 4 , and Li 22 Si 5 . In contrast to the alloying process of Li - Si at high temperature , which follows the phase diagram, the electrochemical reaction of Si with Li at room temperature proceeds via non - equilibrium solid - state amorphization. Limthongkul et al . 31,32 observed a single voltage plateau during the lithiation of Si at room temperature ( Fig. 1 - 5 (b), solid curve ) using X - ray diffraction and HREM . X - ray diffraction 34,38 studies revealed the continuous formation of metastable amorphous Li x Si phases instead of equilibrium intermetallic compounds, which serve as an evidence the electrochemical lithiation at room temperature is a non - equilibrium process. In o ther words, the electrochemical reaction of Si with Li at room temperature is kinetically controlled and since the formation of thermodynamically stable crystalline intermediate phases is circumvented, amorphous phases with lower Gibbs free energy than the reactant but higher than those of the equilibrium crystalline phases are formed. 8 Figure 1 - 6 (a) Schematic diagram of phase evolution upon multiple cycle based on i n s itu XRD of charge/discharge cycles of Li - Si battery (Adapted from reference [ 33 ]. Copyright © Journal of the Electrochemical Society 2007) (b) High - resolution TEM image of the sharp phas e boundary between c - Si and a - Li x Si (Adapted from reference [ 37 ]. Copyright © Physical Review Letters 2011) Further XRD 33 , NMR 35,36 , and TEM analysis 37 captured the reaction process with two - phase lithiation and recrystallization of highly lithiated a - Li x Si phase into c - Li 15 Si 4 , as described in the schematic reaction dia gram of c - Si ( Fig. 1 - 6 (a) ). Chon et al . 37 observed a sharp phase boundary (~ 1nm) separating the crystalline silicon and amorphous lithium silicon using scanning electron microscopy and high - resolution transmission electron microscopy, which further confirmed the two - phase reaction, as shown in Fig . 1 - 6 (b) . To further investigate the local structural evolution in the two - phase region where crystalline Si and amorphous Li x Si coexist, Key et al . 35 used in situ and ex situ NMR techniques, followed by local structure probes and PDF analysi s. 36 Upon initial lithiation stage, Si bonds begins to dissociate and forms small clusters of Si surrounded by Li atoms. As time proceeded, these small Si clusters broke into isolated Si atoms, a reaction indicating complete crystalline - to - amorphous phase transition, which corresponded to th e Li concentration of ~3.4. Further lithiation results in rapid crystallization of a - Li x Si to c - Li 15 Si 4 , which is a characteristic phase transition observed only during electrochemical lithiation of Si at 9 room temperature. 33,36 During delithiation, previously formed metastable c - phase or a - gradually disappears and are replaced by amorphous lithium silicon. NMR studies 35,36 indicate that the two - phase coexists until the terminal phase of the previous lithiation completely disappears and a single amorphous phase emerges towards the end of the delithiation process. Regardless of the initial phase of the Si (c - Si or a - Si), the resulting amorphous Si is different from the initial Si. Recent development of in situ transmission electron microscopy (TEM) techniques enabled researchers to elucidate the local and detailed structural information which previously buried in the averaged properties. 16,43 47 By tracking the structural evolution of single Si nanoparticles or nanowire during lithiation, it was revealed that the short - range reaction and interfacial mobility controls the ra te of phase transformation and determines the volume expansion pattern upon lithiation. Figure 1 - 7 (a) Characteristic volume expansion of c - Si nanopillars along <100>, <110>, and <111> axial orientations upon lithiation (Adapted from reference [ 45 ]. Copyright © Nano Letters 2011 ) (b) High - resolution TEM image and schematic diagram describing the lateral ledge flow along {111} plane (Adapted from reference [ 43 ]. Copyright © Nature Nanotechnology 2012 ) 10 Lee et al . 45 examined the lithiation - induced shape and volume change of c - Si nanopillars with different crystallographic orientations and observed anisotropic expansion. This anisotropic expansion suggested that the rate of the phase transition strong ly depends on the direction of Li insertion, <110> was shown to be the fastest and <111> the slowest. Liu et al . 44 reported anisotropic swelling of Si nano - wires during lithiation using in situ TEM, which further supported the idea that lithiation rate in c - Si is highly anisotropic. Furthermore, the movement of the sharp reaction front was linearly correlated with reaction time, suggesting lithiation is controlled by short - range reactions at the reaction front. Recently, Liu and coworkers 43 revealed the atomic - scale mechanism of the lithiation process is characterized by a layer - by - layer peeling - off of {111} facets using in situ TEM. Taken together with finite el ement modeling, 16 it was determined that the rate of phase transformation and concurrently occurring anisotropic volume expansion upon l ithiation is controlled by the short - range reaction and interfacial mobility. For a - Si, in situ TEM experiments 46,47 revealed unexpected two - phase lithiation behavior, which causes a different mechanism of stress evolution. During the first lithiation process, a sharp phase boundary of nanoscale thickness was observed which separated the a - Li x Si region and unlithiated a - Si region. This indicated that lithiation in a - Si is limited by reaction front mobility , while less Li is required to break the Si - Si bonds in a - LiSi than to disrupt the rigid Si - Si covalent network in c - Si case. Upon lithiation, a - Si expands in an isot ropic manner due to the lack of underlying crystallography. In subsequent lithiation/delithiation cycles, a single - phase mechanism was observed. Experimental observation s introduced above provide important information to understand the electrochemical lithiation/delithiation of Si. However, the inherent limitation of the experiment method s is that they only provide information related to the averaged properties over space and time. For instance, the important Li distribution information upon lithiation/ delithiation cannot be 11 obtained. Also, detailed (local) structural evolutions at different lithiation/delithiation stages are difficult to study since even the most advanced in - situ TEM methods is not sensitive enough to capture the amorphous nature of a - L i x Si. Computational investigation of Li - Si system in electronic and atomic level can provide fundamental knowledge regarding electronic and chemical properties, which information can be further utilized to better understand and predict reaction of Li with Si. 1.1.2 Computational Studies of Li - Si System Figure 1 - 8 Multi - scale modeling and simulation C omputer simulations have become an important tool in essentially all fields of chemistry, condensed matter physics, and material science. 48 In principle, material properties and reaction 12 mechanisms can be describable by quantum mechanics (QM). However, simulations of large system size and dynamics time are impractical since solving the Schr ö dinger equation is computationally too expensive. Classical approximations lead to simplified equations of motion whi ch describes the interaction between atoms and therefore, applicable to much larger system size and longer dynamic calculations without compromising the accuracy of the calculations. At the higher end of the length and time scales, continuum - level methods are widely used which provides insight into elucidating the behavior of matter at the microscale and macroscale. In this thesis, I only focus on the computational simulations on the electronic - and atomic - scale. 1. 1.2. (a) Density Functional Theory Calcu lation of Li - Si Characteristic material properties regarding the thermodynamic, electronic, kinetic, and mechanical properties of the bulk Li - Si system have been extensively studied with first - principle DFT calculations. 49 58 Energetics, charges , diffusivities, and elastic constants of Li x Si at certain compositions were calculated and the relationship between these properties at different Li concentrations was interpreted to obtain electronic and atomic level understanding of the Li - Si system. To determine the structure and stability of Li - Si compounds, Chevrier et al . 52 computed the energies of Li x Si structures at various Li concentrations and revealed the formation energy per Li x Si constantly decrease upon Li insertion and reached its minimum around x = 3.75. Furthermore, Jung and Han 57 determined the crystalline - to - phase transition occurs during the initial stage of lithiation ( x = 0.3) based on the formation energy comparison of a - Li x Si and c - Li x Si. Open - circu it voltage (OCV), which is another important parameter for lithiation experiments, were also computed from energies obtained from DFT calculations. As shown in Fig. 1 - 9 (a) , the computed 13 OCV curve of c - Li x Si and a - Li x Si compounds agreed well with the exper iment which confirmed that essential physics and energetics of Si upon lithiation were successfully captured. 52,53 Figure 1 - 9 (a) Experimentally measured (red curve) and computationally calculated (blue, green, and black curve) v oltage vs. composition of Li x Si . (Adapted from reference [ 52 ]. Copyright © Journal of the Electrochemical Society 2009 ) (b) Net charge of Li and Si in c - Li x Si (LiSi, Li 12 Si 7 , Li 7 Si 3 , Li 13 Si 4 , Li 15 Si 4 , and Li 21 Si 5 ) . (Adapted from reference [ 54 ]. Copyright © Journal of Alloys and Compounds 2010 ) Another fundamental insight that can be obtained from first - principle calculations are the i nformation regarding the charge transfer and electronic structure. Using Bader charge analysis, Chevrier et al . 54 calculated the net charge of the Li and Si atoms at various c - Li x Si , as shown in Fig. 1 - 9 (b) . They revealed regardless of the Li concentration, Li atoms have similar positive net charge of 0.73 e and 0.68 e . In contrast, the net charge of Si atoms decreases as the Li concentration increases, which clearly indicates Si net charge is strongly dependent on the local environment. In other words, decrease in Si net charge is due to the dissociation of Si - Si coval ent bond and formation of Li - Si ionic bonds upon increased Li concentration. To efficiently obtain the a - Li x Si structures while avoiding the computationally expensive Ab Initio Molecular Dynamics (AIMD), Chevrier and Dahn 52,53 successfully developed a protocol 14 based on Li insertion at energetically fav orable sites followed by structural optimization. Using this self - developed lithiation algorithm, they were able to successfully generate a - Li x Si structures which captured the volume expansion and OCV vs composition curve which agreed well with the experim ental values. Chan et al . 51 further improved the Li insertion protocol and performed Li insertion and removal simulation to study the complet e lithiation/delithiation process of c - Si and discovered that Li insertion to c - Si with (110) surface orientations is thermodynamically mo r e favorabl e than (100) and (111). For mechanical properties, Shenoy et al . 49 calculated the elastic modulus of c - Li x Si and a - Li x Si c - Li x Si and a - Li x Si decrease linearly with increasing Li concentration, which indicated a softening effect as Li x Si approached the Li - rich phases. Elastic softening phenomena upon increase in Li c oncentration is due to the breakage of strong covalent Si - Si bonds being replaced by weak ionic Li - Si bonds. Finally, diffusion kinetics and local structural evolution upon lithiation was also computed using ab initio molecular dynamics (AIMD) simulations. Johari et al . 56 investigated the Li - Si mixing mec hanism upon lithiation from AIMD using a slab model with 1D diffusion and revealed that Li atoms convert the conventional 6 - node - rings by interrupting the Si - Si covalent bonds. The structural evolution continued with time by forming temporary structures, w hich eventually evolved into isolated Si atoms and Si - Si dumbbells, as shown in Fig. 1 - 10 (b) - (c) . Furthermore, diffusivity calculations of Li and Si suggested that both Li and Si diffuse faster in a - Si than in c - Si and room temperature Li diffusivities, obtained by extrapolating high temperature results, is in good agreement with experimentally measured Li diffusivities. Recently, Wang et al . 58 revealed Li diffus ion increases with Li concentration with a linear trend based on calculated Li diffusivity at four different concentrations. 15 Figure 1 - 10 (a) Representative structural snapshots at various lithiation stages of c - Si (top) and a - Si (a - Si) at 1200K. Cyan and purple color spheres in the figure represent Si and Li atoms, respectively (b) Representation of structures form during lithiation (c) Structu ral statistics with time upon for lithiation of c - Si . (Adapted from reference [ 56 ]. Copyright © Nano Lett ers 2011 ) Although first - principle calculations of Li - Si system provides valuable fundamental insights of chemical, electronical, and mechanical properties, most of them are from static property calculation of Li x Si compounds at equilibrium. Ab initio MD simulations describe the short - range local structural change characterized by breakage of Si clusters into isolated Si atoms. However, the simulation size is limited to ~ 100 atoms due to the computational cost, thus the information regarding the movement of reaction front and long - range structural evolutions are buried inside the immediate collapse (lithiation) upon mixing. 56 In other words, within the scope of first - principle calculations, it is difficult to simultaneously track and correlate the chemical and structural e volution upon lithiation and delithiation. Atomistic simulation, which approximates 16 atoms as classical particles and replace the expensive quantum calculations with interactions between atoms, is a great alternative to study the dynamic evolution of large structures in time. 1.1.2. ( b ) Molecular Dynamics Simulation of Li - Si Molecular Dynamics (MD) simulation, also known as atomistic simulations, can simulate over scales relevant to nanometers and nanoseconds . Therefore, it is considered as a promising alternative method to study the dynamic properties change but the usage was restricted in Li - Si system due to the lack of accurate interatomic potentials to describe the chemical reactions between Li and Si. Recently, van Duin and coworkers 59 61 developed a reactive fore field (ReaxFF) for Li - Si system, which accurately describes the bond breaking and bond formation processes, thus providing a set of reliable properties for Li x Si alloys including chemical reactions, volume expansion, and open - cel l voltage. Figure 1 - 11 (a) Structural snapshot representing the lithiation of a - Si and c - Si with (100), (110), and (111) surface orientations after 200 ps at 1200K using ReaxFF - MD simulations (Adapted from reference [ 62 ]. Copyright © The Jo urnal of Physical Chemistry C 2014) (b) Structural snapshot representing the delithiation of lithiated Si NW with MD equilibration time of 250 ps (red) and 500 ps (black) for each delithiation steps (Adapted from reference [ 63 ]. Copyright © The Journal of Physical Chemistry C 2015) 17 Motivated by in situ TEM experiments, which revealed the phase boundary, Kim et al . 62 investigated the formation and propagation of phase boundary for c - Si with different orientations. They revealed the phase boundary depends on the orientation of c - Si, where the location of the (111) plane governs the rate of crystal - to - amorphous phase transformation and the thickness of the phase boundary. Ostadhossein et al . 64 also studied the Li insertion process into c - Si nano - wires using ReaxFF and revealed the atomistic mechanism of the crystalline - to - amorphous phase transf ormation. They also demonstrated that Li diffusion - induced compressive stress can slow down the lithiation process. The consistency with experiments validated the ability of ReaxFF to capture the room - temperature electrochemical reactions of the Li - Si syst em. Jung et al. 63 simulated the delithiation process by removing the Li atoms from the surface of a - Li x Si to generate a Li concentration gradient, which served as the driving force for delithiation. They revealed the formation of c - Si nuclei in the delithiated a - Li x Si matrix and demonstrated that the volume of delithiat ed a - Li x Si is larger than the original Si volume. However, removing all the Li from the natural in terms of volume contraction, structural evolution, Li diffusion in the Li - Si system. These experiments and computational studies have investigated the reaction of Si upon lithiation/delithiation, but there are still important information missing regarding the lithiation rate and the origin of the irreversible capacity lo ss. Lithiation rate is one of the most important factor s that determines the performance of Li - ion battery since it significantly impacts the stress generation and determines the rate performance. In order to improve the rate performance by accelerating t he lithiation rate, it is crucial to elucidate the rate - limiting factor upon lithiation. Also, the effect of Li concentration on Li diffusivity, which affects the lithiation rate on different stages of lithiation, remains unclear. 18 The origin of the irreve rsible capacity loss that are inherent at atomic - scale is another key factor missing to understand the high irreversible capacity loss. Typically, initial irreversible capacity loss is responsible for the poor capacity retention in Si anode. In other words , besides the consumption of Li atoms in SEI layer formation and other side reactions, the irreversible capacity loss is mainly due to the chemical and structural evolution during the delithiation. So far, simulation of delithiation is in its infancy level since an effect ive delithiation procedure to a systematic delithiation algorithm needs to be developed and utilized to simulate the delithiation process which can simultaneously tr ack and correlate the irreversible chemical and structural evolution with rate effects. 1.2 Motivation and Thesis Outline The goal of this thesis is to use electronic and atomic - scale simulations to obtain atomic insights on the lithiation mechanism to accelerate rate performance, elucidate the irreversible changes that are inherent at atomic - scale upon delithiation with rate effects for Li - Si system. Furthermore, we will evaluat e the reactive force field for application to SiO system and lay down th e ground work to design extensive training set of Li - Si system for machine - learning potential development. In Chapter 2, computational tools and methods are introduced. Basics of density functional theory and molecular dynamics with force field are explai ned in detail. Chapter 3 presents the ReaxFF - based molecular dynamics to elucidate the rate - limiting factor upon lithiation of Si for accelerated lithiation rate. Fundamental insights on lithiation dynamics of c - Si with different orientations and a - Si wer e extensively studied and the effect of Li 19 concentration on Li diffusivity were revealed. Also, based on the determination of the rate - limiting factor, methods to enhance the lithiation rates are introduced. Chapter 4 focuse s on understanding the irrevers ible changes that are inherent in atomic - upon delithiation, we first developed a continuous reactive molecular dynamics delithiation algorithm, with well - cont rolled chemical potential gradient driving force and delithiation rate. With this new systematic delithiation algorithm, the fundamental reasons behind initial irreversible capacity loss was investigated by analyzing the relationship between the depth of d ischarge and corresponding volume and structural changes at different rates. Furthermore, the effect of irreversible structural changes on subsequent lithiation processes was also studied. In Chapter 5 , the reactive force field developed for Li/Al/Si/O/H s ystem are evaluated for application to simulate lithiation/delithiation of SiO system. The phase stabilities of SiO x structures are computed against DFT calculations and design of the training set for modified Li/Si/O parameters are suggested. Chapter 6 p resents the development of accurate and extensive training set for the next - generation machine - learning potentials for Li - Si system. Based on Ab Initio Molecular Dynamics and DFT optimization procedure, systematic strategy to generate Li - Si training set wi th minimum human intervention are developed and the quality of the training set is also evaluated. Finally, Chapter 7 presents a summary of the research work and discusses future research directio n. 20 Chapter 2 Overview of Computational Methods 2.1 Energy Calculation using DFT and Force Fields 2.1.1 Density Functional Theory First principle calculations describe the interaction of electrons by solving Sch r ö dinger equation. Born - Oppenheimer approximation 65 states that the motion of nuclei and electrons can be separated since electrons move much faster than nuclei, thus nuclei can be treated as station ary. Based on Born - Oppenheimer approximation , atomic positions are treated to be fixed and wave functions are used to solve the Schr ö dinger equation which describe the motion of electrons. The time - independent Schr ö dinger can be expressed as (2 - 1) where H is the Hamiltonian operator, E is the energy operator, and is the wave function. In case where the Hamiltonian has a simple form, such as hydrogen system, Schr ö dinger equation can be exactly solved. However, it is impossible to solve Schr ö dinger equation with multiple nuclei and electrons. For complicated systems with multiple nuclei and electrons, Schr ö dinger equation can be expressed as (2 - 2) where m and N refers to the mass of electron and total number of electrons. The three terms in brackets refer to the energies corresponding to, in order, kinetic electrons, interaction between each electron and atomic nuclei, and electron - electron interactions. Among various methods to solve Schr ö dinger equation with multiple nuclei and electrons , density functional theory (DFT) is one 21 of the most widely used method due to its efficiency . D FT 66 , first proposed by Kohn and Hohenberg in 1964, is based on two key theorems: 1. The ground - state energy from Schr ö is determined by a unique functional of the electron density (2 - 3) 2. The correct electron density, which describes the ground - state, minimizes the energy of the overall system Based on th ese theorems , Kohn and Sham developed DFT which reduces the intractable interactions of many electrons to a tractable functional of the electron density. Using DFT, time - independent Schr ö dinger equation ( eqn. 2 - 1 ) can be describe by a wave function (2 - 4) where the first term is defined as the kinetic energy of electrons, V(r) is the sum of electron interaction, V H (r) 66 is the electron - electron interaction and the self - coulombic interaction energy. Exchange - correlation contributions, V XC (r) 66 , describes the interaction between electrons which is difficult to capture in DFT since DFT treats electrons as a functional of electron density. V XC (r) 66 is defined as a function derivative of XC energy (2 - 5) The exact form of exchange - correlation term is unknown. Therefore, various approximate functionals of electron density, such as local density approximation (LDA), generalized gradient approximation (GGA) , are applied to describe this term. All DFT calculat ions performed in this 22 thesis uses Perdew - Burke - Ernzerhof (PBE) functional (a class of GGA), which utilizes both electron density and its gradient. In DFT, plane wave basis sets are used to represent the electronic wave functions; (2 - 6) where is the normalization factor (volume of the box), G is the reciprocal lattice vector and r is the real space p osition . For practical application, we must basis sets must be truncated since DFT c alculation with infinite plane - wave basis sets are difficult to perform. In this purpose, I utilized the cut - off energy in our DFT calculations and kept it constant throughout different calculations (2 - 7) Also, in our simulation (periodic system), integrals over the first Brillouin Zone in reciprocal space was used. Rather than integrating all the possible K - points, I carefully selected representative points (KPOINTS) with convergence tes t, to maintain high computational efficiency. 2.1.2 Force Fields 2.1.2. (a) Introduction Force Field is a collection of functional forms and parameter sets, which is used to determine the energetics of large number of atoms (beyond the scope of DFT) at a reasonable computational cost. Based on Born - Oppenheimer approximation , computationally expensive electronic calculations are ignored and only the motions of the nuclei are utilized to describe the energ y of the system. Additivity and transferability are two characteristics which defines a force 23 field. 69 71 Additivity means that the system energy can be expressed as sum of different energy contributions corresponding to bonded and non - bonded interactions. Transferability indicates the ability of the force field to use a force field developed for a specific atomic environment to descri be a much larger atomic system with similar chemical environments. Although f orce f ield can efficiently capture the energetics based on atomic interactions, it is impossible to correctly translate all the quantum mechanical effects into one optimal set of functionals and parameters. Therefore, based on the system (covalent, ionic, metallic, etc.) of interest, different types of force fields with unique functional forms and parameters (empirical or from first - principle calculation) are designed. Classical f o rce f ield 69 71 and Reactive f orce f ield (ReaxFF) 61,72 74 are two main type of f orce f ield which is widely used. Therefore, in the following section, Classical and Reactive f orce f ields are introduced. 2.1.2. (b) Classical Force Field Classical f orce f ields, such as AMBER, CHARMM, and GROMOS, are mainly used in the field of chemistry and biochemistry, which focus on describing the covalently bonded system. Most classical force fiel ds can be expressed as potential energy terms corresponding to deformation of bond and angles, rotation of dihedral angles, van der W aals interactions , and electrostatic interactions, 69 71 as shown in Fig. 2 - 1 . The first three terms in the equation above represent covalent ly bonded terms , which corresponding energies are described with a simple harmonic potential of a force constant, equilibrium, and actual distance, angle, and torsion. First, s econd , and third term represents the energy contribution of deformation of bond, angle, and rotation of dihedral angle (torsion). The fourth and fifth term in the equation repre sent the non - bonded terms . The fourth term represent the contribution of van der Waals interaction, which 24 includes both attractive and repulsive term. The fifth term describes the Coulomb ic potential (electrostatic interaction) and are calculated by using charge equilibration (QEq) method. 75 Figure 2 - 1 Description of energy terms for classical force fields. Each number represents (1) B ond, (2) Angle, (3) Dihedral, (4) van der Waals, and (5) Electrostatic contribution Calculation involving classical force field successfully describe the non - reactive interactions near equilibrium. However, rigid functional form (harmonic) employed in the classical force field are inadequate for modelling chemic al reactions with change in connectivity. For example, simple harmonic potential is applied to describe the angle and torsion interactions 25 captured or simulate d using classical force field. Simulating chemical reactions can be achieved 2.1.2. (c) Reactive Force Field Figure 2 - 2 (a) Overview of the ReaxFF total energy com ponents 61 (b) Interato m ic distance dependency of th e carbon - carbon bond order 72 Energy contribution of ReaxFF can be expressed as shown in E qn. 2 - 9 , where each term represents bond deformation , over - coordination penalty , under - coordination stability, lone - pair, valence angle , torsion, and non - bonding v an der Waals and Coulombic energies, respectively. 60,61 Different from classical force field, ReaxFF utilizes a bond - order dependent scheme to model the interactions between different atoms, as describe in the schematic in Fig. 2 - 2 (a) . By converting the bond distance to bond order, ReaxFF is able to dynamically describe the atomic interaction without predefined r eactive sites. 61 The relationship between the bond distance and bond order are illustrated in E qn. 2 - 1 0 , where each exponential term represent the contribution from single, double, and triple bonds; 26 (2 - 10) where and represent equilibrium bond distance of single, double, and triple bonds. Each p term represents parameters obtained from first - principle calculations to successfully capture bond strength and corr esponding energies for species that are r ij apart. Fig. 2 - 2 (b) illustrates an example of unique ReaxFF bond order scheme for carbon - carbon interaction 72 , which highlights a smooth transition from nonbonded to single, double, and triple bonded system due to bond order dependent scheme. In terms of non - bonded interactions, ReaxFF uses electronegativity and hardness parameter to calculate the charge transfer within the system. Based on EEM 76 and QEq 75 meth ods, charge s are at each time step are calculated using geometry - dependent scheme, and further utilized to determine the C oulombic interactions. Also, different from classical force field which only calculates the non - bonded interactions between atoms that are not directly connected, ReaxFF calculates C oulombic and van der Waals interactions between all the atom pairs. To prevent excessive repulsion and attraction, ReaxFF employs a shielded term and corrects the interactions at short distance. 2.2 Mo lecular Dynamics 2.2.1 Ab Initio MD and Force Field based MD Molecular Dynamics (MD) is a computer simulation method to study the thermodynamics and dynamic properties of a system by tracing the motion of atoms based on statistical mechanics. MD cons ider atoms as classical particles and determines the dynamic evolution by numerically 27 (2 - 1 1 ) where F i is the force ac ting on an atom, m i is the mass , and a i is the acceleration of atom i . The potential energies and forces between atoms are calculated via DFT o r force fields . In this thesis, the first is referred as Ab Initio MD (AIMD) and later is referred as classical MD (with classical force field) or reactive MD (with reactive force field). In Ab initio MD, instead of using a prescribed potential, one solves the interatomic forces at a given time instant as follows. From a quantum - mechanical perspective, the system at a fixed time can be parametrized in terms of the coordinates of the nuclei and the relevant electrons. Based on Born - Oppenheimer approximation, the nuclei can be considered fixed and the time - independent Schrödinger equation can be written as the many - body wave function of the electr ons. This Schrödinger equation is then solved using (time - independent) DFT to obtain the energy. The energy is then considered to be a function of the nuclear coordinates that were fixed earlier, and it can thus act as the interatomic potential that is needed t o compute the forces in Newton's equation of motion for the nuclei. So, by computing the gradients of the DFT energy at this fixed point with respect to the nuclear coordinates, forces are obtained and the nuclei are moved according to get to the next time step. The DFT process is then repeated with these new nuclear coordinates. The difference between Ab Initio MD and f orce f of motion are determined. In the standard MD approach, th e instantaneous force on each atom is calculated as a gradient of a prescribed interatomic potential function (or, in other words, force field), which is function of the atomic coordinates that you can equally well regard as the coordinates of their nuclei . By avoiding computationally expensive electronic calculation with efficient force field, much larger system with longer dynamic simulation can be perform. 28 Ab initio MD and f orce f ield based MD approaches differ significantly in terms of the ability and computational cost due to the calculation method. Ab initio MD calculates the forces by on - the - fly from accurate electronic structure calculations, which intrinsically captures the b onding, charge transfer, polarization, and many - body effects. However, simulation with only small system sizes and short simulation time can be utilized due to significant computational cost. In f orce f ield based MD, one of the most challenging aspects is the accuracy of the calculation correctly translate all the quantum mechanical effects into one optimal set of functionals and parameters . Also, force fields ca n only describe the environment which it learned from and it is difficult to extrapolate to configurational space not included in the training set. Despite the limitation rising from force fields, MD with force field can simulate much larger system size wi th longer simulation time, thus providing valuable information regarding the phase, structures, and dynamics at a reasonable accuracy to compromise. 2.2.2 Equation of Motions and Thermodynamic Ensembles Figure 2 - 3 Illustration of molecular dynamics algorithm 29 F low chart in Fig. 2 - 3 describes the MD algorithm. Initially, the positions and velocities of atoms are specified , which propagates with a finite time interval using numerical integrators, such as Verlet algorithm. 77 of each particles at different time in the system can be determined . As the particle moves, averaged properties including thermodynamic information, phase transition, structural e volution, and diffusion dynamics are obtained by analyzing the movement of the atoms (trajectories). Various thermodynamic conditions can be simulated by controlling three thermodynamic state variables: Pressure (P), Volume (V), and Temperature (T). NPT ensemble, also known as isothermal - isobaric ensemble, represents a simulation condition where the pressure and temperature are controlled to be constant. It is mainly used to reproduce the experimental condition, where correct pressure and densities a re important. NVT ensemble (canonical ensemble) is used to simulate a thermodynamic condition with constant volume. Similarly, for constant volume and total energy, NVE ensemble (microcanonical ensemble) is utilized. 30 Chapter 3 Vacancies in Si Can Improve the Concentration - Dependent Lithiation Rate 3.1 Summary The study of lithiation dynamics is important since it affects both stress generation and rate performance of electrodes for Li - ion batteries. This topic becomes more crucial for Si anodes because its high capacity is accompanied by dramatic volume and structural changes, which lead to mechanical fracture, capacity loss, and limited cycle life. In order to provide fundamental insights into the lithiation dynamics, determine the rate - limiting process of lithiation, and investigate the effect of concentration on Li diffusivity, molecular dynamics (MD) along with reactive force field (ReaxFF) was used to simulate the lithiation process of both amorphous and crystalline Si electrodes. The local lithiatio n concentration evolution shows that lithiation dynamics can be characterized as two stages: an initial mixing stage followed by subsequent random walk diffusion stage. The Li diffusion is demonstrated to be concentration dependent as Li diffuses faster with higher Li concentration, opposite to many intercalation compounds. The degree of Li diffusivity increment with respect to Li concentration increases dramatically around Li 0.8 Si. This relationship provides an underlying reason for the experimentally ob served two - phase lithiation in both c - Si and a - Si. Furthermore, it is found that the lithiation rate during the initial mixing stage increases exponentially with vacancy concentrations in Si. This relationship reveals that the Si - Si bond breaking is the ra te - limiting factor for Si lithiation. This chapter is reproduced from the work published as: Kwang Jin Kim, Yue Qi, - Dependent Lithiation Rate: Molecular Dynamics 2015, 119, 24265 - 24275. 31 3.2 Introduction Si is considered as a promising anode material due to its exceptionally high the oretical specific capacity (3750 mAh/g) and abundancy, which is promising in terms of its availability for future usage. Despite its high specific capacity, Si is limited in its application due to its 300 % volume expansion and massive structural change during lithiation and delithiation, caused by the large number of Li reacting with the host material. This leads to mechanical fra cture and disconnection between the particles, which contributes to the irreversible capacity loss and limited cycle life. 7,9 12 It is important to understand the fundamental aspect of lithiation/delithiaton dynamics since the stress generation and rate performance are significantly affected by the diffusion of Li and the host materials in alloy forming electrodes. During the lithia tion process, Si undergoes enormous structural changes accompanied by significant changes in electronic and mechanical properties. Experiments, including X - ray diffraction (XRD), nuclear magnetic resonance (NMR), and acoustic emission have traced and reve aled the evolution of averaged chemical bonding and structural change during Si cycling. 31 39 It has been shown that electrochemical lithiation of crystalline S i (c - Si) at room temperature involves solid - state amorphization. Due to this phase transformation, c - Si particles have shown more fractures during amorphization than during subsequent cycling. To avoid initial cracking due to amorphization, amorphous Si (a - Si) electrodes are preferable in practical battery applications. Thus, understanding the difference between the lithiation dynamics in a - Si and the lithiation dynamics in c - Si is not only scientifically interesting but also practically important. 32 Recent development of in situ transmission electron microscopy (TEM) techniques which tracked the structural evolution of single Si nano - particles or nano - wires during lithiation, revealed that the rate of phase transformation and concurrently occurring anisotropic volume expansion upon lithiation is controlled by the short - range reaction and interfacial mobilit y . Similarly, the initial lithiation of a - Si proceeds via two - phase lithiation wh ich highlights the reaction front limited diffusion. 16,43 47 In the field of theoretical research, initial simulations on the lithiation process of both c - Si and a - Si relied on first - principles calcu lations, which characterized and predicted diffusion energy barriers, formation energies, and elastic properties. 49 58 More specifically, ab initio molecular dynamics (AIMD) simulations were used to gain fundamental insights on the lithiation dynamics at the atomic level. 56,58 Johari et al . 56 investigated the Li - Si mixing mechanism upon lithiation of c - Si and a - Si from AIMD using a slab model with 1D diffusion and revealed that structural evolution proceeds by Li atoms interrupting the Si - Si covalent bonds and eventually evolves into isolated Si atoms and Si - Si dumbbells. Further radial distribution functions and pair distribution function analysis indicate AIMD successfully capture the lithiation - induced amo rphization. Diffusivity calculation of Li and Si suggested that both Li and Si diffuse faster in a - Si than in c - Si. To further investigate the lithiation behavior with larger system size and longer dynamics, Kim et al . 62 investigated the formation and propagation of phase boundary for c - Si with different orientations using ReaxFF - based MD simulations. They revealed the phase boundary depends on the orientation of c - Si, where the location of the (111) plane governs the rate of crystal - to - amorphous phase transformation and the thickness of the phase boundary. 62 Ostadhossein et al . 78 also studied the Li insertion process into c - Si nano - wires using ReaxFF and revealed the atomistic 33 mechanis m of the crystalline - to - amorphous phase transformation and demonstrated Li diffusion - induced compressive stress can slow down the lithiation process. Although many experiments and computational studies have investigated the lithiation process of Si, the r ate - limiting factor of this process has not yet been determined. In this work, I performed MD simulations with ReaxFF to examine lithiation dynamics in c - Si with different surface orientations and a - Si at t he atomic - scale. Furthermore, I studied how to acc elerate the lithiation rate by determining the rate - limiting factor in Li - Si system. We also investigated the effect of Li concentrations on Li diffusivity, which demonstrates the decisive role of Li concentration on the experimentally observed two - phase m echanism upon lithiation. 3.3 Simulation Methods 3.3.1 Reactive Force Field for Li - Si System To perform molecular dynamics with reactive force field, reactive force field developed for Li - Si system was employed. The pioneer work in development of reactive force field for Li - Si system was performed by van Duin et al ., who developed the ReaxFF for S i and Si - Oxide systems. 73 In this work, parameters were trained against a set of DFT - calculated d ata for a wide variety of well - known condensed phases and clusters , including equation of states (total energy versus volume) of Si (sc, d iamond, ), SiO 2 ( , trydimite, coesite, stishovite), dissociation energies of single and double bonds of Si - Si and Si - O in Si/O/H clusters, energies of various Si/O/H clusters as a function of valence angles Si - O - Si, O - Si - O, and Si - Si - Si and distortion energies of rings of Si/O/H clusters. Fan et al ., 59 further opti mized the parameters against a training set from DFT calculations which contained a collection of energies, geometries, and charges relevant to the Li - Si system including equation of states for the 34 body - centered cubic (bcc), face - centered cubic (fcc), and hexagonal - close - packed (hcp) phases of Li and crystalline LiSi, Li 12 Si 7 , Li 13 Si 4 , and Li 15 Si 4 , as shown in Fig . 3 - 1 . Figure 3 - 1 Energy versus volume of (a) Various Li crystals and (b) crystalline LiSi and Li 13 Si 4 predicted by DFT and ReaxFF calculation (Adapted from reference [ 59 ]. Copyright © Modelling and Si mulation in Materials Science and Engineering 2013) A total 10 parameters were fitted to a training set containing 142 data points using a successive single - parameter search method with multiple cycles to account for parameter correlation. The performance of ReaxFF for Li - Si was evaluated by open - cell voltage, volume results. Therefore, in this study, the ReaxFF developed for Li - Si system was employed to simulate the l ithiaton process and determine the rate - limiting factor to accelerated lithiation rate and effect of Li co ncentration on Li diffusivity. 3.3. 2 Molecular Dynamics Simulations To study the lithiation dynamics, ReaxFF - based MD simulations implemented in the Large - scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) were performed. It is well known that Si anodes undergo massive volume expansion and crystal - to - amorphous ph ase 35 transformation upon lithiation. Therefore, to obtain the optimized coordinates and lattice parameters, which incorporate the effect of volume expansion and phase transformation, density of a - LiSi structure at room temperature was first calculated and u sed as a reference to construct the simulation cell size. The c - LiSi was first melted at a high temperature (2500 K) and then quenched to room temperature using NPT ensemble while the ratio of Li and Si was kept constant. The obtained density of a - LiSi at 300 K was 1.91 g/cm 3 . System Simulation cell (Å) # of atoms Time required to reach uniform concentration (ps) x y z 900K 1200K 1500K Dense Si Dense Si Si with 5 % vacancies Dense Si Si with 5 % vacancies a - Si 28.329 28.329 45.438 2400 800 720 520 460 c - Si (100) 26.881 26.881 57.700 2744 >1 ns >1 ns >1 ns 740 640 c - Si (110) 27.154 26.881 52.457 2520 >1 ns >1 ns >1 ns 660 580 c - Si (111) 26.605 26.881 57.108 2688 >1 ns >1 ns >1 ns 780 680 Table 3 - 1 Summary of time required to reach fully lithiated stage, together with simulation information of amorphous Si and crystalline Si with (100), (110), and (111) surface orientations Once the targeted density for the simulation cell was determined, slab structures of a - Si and c - Si with surface orientations (100), (110), and (111) were prepared, and Li atoms were packed into the simulation cell as amorphous at both sides of the Si slab. Dimensions for simulation cells are listed i n Table 3 - 1 . 3D periodic boundary conditions were implemented to mimic an infinitely large Si slab sandwiched between two Li slabs, with two Li/Si interfaces included in each simulation cell. Therefore, the main lithiation direction is perpendicular to the Li/Si interfaces. In order to simulate the lithiation process, Li and Si atoms were initially completely separated and the negative heat of formation drove Li atoms to mix with c - Si and a - Si during the MD simulations. MD simulations of NVT ensemble with t he Nosé - Hoover thermostat and the velocity verlet integration algorithm with a time step of 1.0 fs were applied to simulate the lithiation process, 36 especially the initial mixing process and the subsequent longtime random walk diffusion process. Within the NVT ensemble, the initial configurations were allowed to locally optimize their geometry by minimizing the energy for approximately 2000 MD time steps, corresponding to 2 ps. Then, lithiation dynamics simulations were performed for 1 ns in both a - Si and c - Si. MD simulations were performed at high temperatures (900 K ~ 1500 K) to accelerate the reactions, therefore reducing the simulation time. These temperatures are much lower than the melting temperature of Si, and thus will not change the reaction or diff usion mechanisms and allow us to study the chemical reactions on time scales relevant for MD simulations. 55,56,79,80 To determine the rate - limiting factor upon lithiation, 5 % random vacancies in Si and Li were created separately and lithiathion dynamics were compared. Furthermore, the lithiation dynamics was investigated in more detail for c - Si with 2 %, 4 %, 6 %, 8 %, and 10 % vacancies, in order to reveal the effect of vacancy concentration on lithiation rate. In addition to slab models, bulk structures of a - Li x Si with x ranging from 0.1 ~ 4.4 were prepared using similar heating and quenching methods. NVT dynamics w ere performed at temperatures ranging from 900 K~1500 K and the diffusivities of Li and Si atoms during subsequent diffusion period were computed (the determination of the subsequent diffusion period will be discussed specifically for each case in S ection 3.4 .4 ). 3.3. 3 Local C oncentration and D iffusion P roperties A nalysis In order to investigate the characteristic lithiation dynamics and track the evolution of Li concentrations, local concentrations of successive stages of lithiation were computed. The information regarding local concentrations was obtained by dividing the simulation cell length corresponding to the lithiation direction into 21 bins with equal size (~2 Å) and calculating the 37 ratio of Li and Si atoms corresponding to the specific bins. For Li x Si, the Li concentration, , was defined. The local y was tracked for each bin at different time steps to reveal the evolution of Li concentrations. In typical MD simulations, mean squared displacement (MSD) is the most common measure of the average distance traveled by random motions in a time interval. Base d on MSD, the diffusion coefficient at a given temperature was also calculated using the Einstein relation: (3 - 1) where r i (t) is the atomic positions at time t and q i is a numerical constant that depends on dimensionality. In this work, q i value is 6 , which represents three - dimensional diffusion. During the initial mixing period, the motion of Li and Si are not random. So the MSD should only be calculated after the s ystem evolves into subsequent diffusion period, where Li diffusion is . Once the system reaches the subsequent diffusion period, the averaged MSD and diffusivity were calculated. Similar to local concentrations, local mean square displacements (LMSD) and diffusivities were also defined. The measures the average atomic diffusion distance during time interval of in each bin, as (3 - 2) where is the total number of atoms in the given bin. Since the atoms can move in and ou t of the bins, the f inal atomic position s at time w ere used to assign atoms into different bins. 38 3.4 Results and Discussion Figure 3 - 2 (a) ~ (e) Structural snapshots and local concentrations profile at successive stages of lithiation of a - Si and (f) ~ (j) c - Si with (100) surface orientation. All snapshots are taken at 1200 K. In the figure, spheres in blue and green color represent Si and Li atoms, respectively 39 3.4.1 How Lithiation Proceeds in a - Si To track the lithiation dynamics of a - Si, a series of MD simulations at elevated temperatures (900 K, 1200 K, and 1500 K) were performed. R epresentative lithiation dynamics are illustrated in Fig. 3 - 2 . The snapshots at 0 ps, 1 0 ps, 80 0 ps and 1 ns at 1200 K are shown in Fig . 3 - 2 ( a ) to ( d ) . In order to accurately track the lithiation front, local concentrations for these four snapshots were compared in Fig . 3 - 2 ( e ) . At time t = 0, a sharp interface separates the Li and a - Si slabs. The spontaneously lithiation process is driven by the negative heat of formation of Li x Si. The mixing proceeds as Li atoms move into the Si slab and Si expands outward, resulting in increasing Li concentration in the center of the Si slab. At 10 ps, the center of the Si slab is still not lithiated (y = 0). The Li concentration in the Si slab increases as a function of time. At 800 ps, the concentration distribution in the a - Si is uniform, indicating it is completely mixed. Little change in local Li c oncentration is observed after 800 ps until 1 ns. By tracking the local concentration distribution, we determine the fully lithiation time as the time required to reach the uniform Li concentration through the simulation cells. At 1200 K, it requires 800 p s to fully lithiate the a - Si and it becomes faster at 1500 K, which only takes 520 ps. At 900 K, the structure did not reach its fully lithiated stage within our simulation time (>1 ns). 3.4.2 How Lithiation Proceeds in c - Si Structural configurations during Li insertion into c - Si with (1 00 ) surface orientations at 1200 K are shown in Fig. 3 - 2 ( f ) to ( j ) . As shown in Fig. 3 - 2 ( f ) and (j) , lithiation into the initially c - Si is still in progress after 1 ns, whereas the initially a - Si is completely mixed after 800 ps. This clearly indicates that reaction is more active in a - Si, as expected due to the less ordered Si - Si covalent bonds, which re sult in easy disso ciation upon lithiation. 40 We have observed the orientation dependent lithiation dynamics in c - Si, similarly to the anisotropic effect discussed in the literature. 16,43 45,81 It can be seen that lithiation behavior in c - Si is controlled by hopping diffusion of Li atoms between the tetrahedral sites, which has been proven to be the most stable position in c - Si upon Li insertion. 82 T o verify whether ReaxFF successfully captures the preferential Li insertion into tetrahedral sites (T d ) in c - Si, we calculated the bi nding energies of Li atom in c - Si at four different potential Li atom insertion sites including tetrahedral, hexagonal, bond - center, and next nearest neighbor sites. Among all four sites, the tetrahedral site is the most stable position for Li atoms. It is 0.65 eV lower in energy compared to the hexagonal site (the second energetically favorable position), which is in good agreement with the value obtained from first - principles calculations (0.55 eV). 83,84 Furthermo re, the calculated energy barrier for a Li atom to hop between two tetrahedral sites in the c - Si using ReaxFF is also in good agreement with that obtained from first principles calculations. 62,85 These results confirmed that ReaxFF can accurately capture the energetics of Li insertion and hopping diffusion in c - Si. This lithiation pattern continues as Si gradually loses its crystalline structure and becomes amorphous. At 1 ns, although some local order can be o bserved in the structure, the overall radial distribution function shows a typical amorphous structure character. As lithiation proceeds, the Si - Si covalent bonds with all nearby tetrahedral sites occupied by Li atoms become weakened and eventually break. This phenomen on is due to both mechanical swelling and the charge transfer from Li to Si. The charge transferred to Si fills up the antibonding sp 3 state of Si, subsequently weakens, and eventually breaks the corresponding Si - Si covalent bonds. 86 Similar lithiation mechanism (Li occupying the first available tetrahedral sites) occurs in c - Si (111), however the location of first available tetrahedral sites are normal to the lithiation direction, thus creating a layer of Li atoms near the Li/Si inte rface. Accumulation of Li atoms results in a high local 41 concentration of Li atoms between the adjacent (111) planes, which causes breakage of Si - Si covalent bonds between the (111) bilayers. This results in a ledge mechanism, which is characterized by the peeling off of (111) planes. The layer - by - layer cleavage of (111) planes upon lithiation clearly explains the atomic mechanism of crystalline - to - amorphous phase transformation, which supports the in situ TEM observation. 43 The amount of time required for each structure to become fully lithiated is summarized in Table 3 - 1 . For all of the cases, lithiation in a - Si proceeds faster than in c - Si. Among c - Si orientations, c - Si with surface orientation (110) reaches the fully lithiated stage the fastest due to fast Li ion diffusion channel along <110> direction, consistent with experiments. 45 3.4.3 Si Vacancy Generation Accelerates Lithiation Dynamics More importantly, the unanswered questions are what the rate - limiting factor is during the lithiation of the Si anode and how to accelerate the lithiatio n process. As shown in Fig. 3 - 3 (a) - (c) , the lithiation rate is accelerated when vacancies are introduced in Si, whereas vacancies in Li have negligible impact on the lithiation dynamics. Fig. 3 - 3 (d) represents the location of the reaction fronts of systems with no vacancies, 5 % Li vacancies, and 5 % Si vacancies after 10 ps at 1200 K. The location of the reaction front is marked by the furthest Si bins with non - zero Li concentration. As clearly ind icated in the plot, the location of the reaction front is further into the Si slab when Si vacancies are introduced, which indicates the lithiation rate is dramatically increased by Si vacancies. In comparison, the presence of vacancies in Li does not chan ge the lithiation rate at all. 42 Figure 3 - 3 S tructural snapshots of c - Si with (100) surface orientations with (a) no vacancy, (b) 5 % Li vacancy, (c) and 5 % Si vacancy after a reaction time of 10 ps at 120 0 K. In the figure, spheres in blue and green color represent Si and Li atoms, respectively (d) The concentration profiles of the three snapshots. (e) Movement of reaction fronts with and without Si vacancies with respect to time at 1200 K To further illustrate the effect of vacancies on lithiation dynamics, the locations of lithiation reaction fronts with and without Si vacancies were tracked as a function of time, as shown in Fig. 3 - 3 (e) . The lithiation rate is four times faster with 5 % Si vacanci es. Additionally, a clear linear trend is observed in both cases indicating that short - range processes at the reaction front control the lithiation dynamics, agreeing well with in situ TEM experiments in both c - Si and a - Si. 44,47 Even though our previous calculations suggested that Si atoms remain relatively stationary during lithiation, 56 the empty space generated by Si vacancies facilitates local rearrangement of Si atoms to readily accept Li insertion. Furthermore, our results indicate that the movement of Si 43 atoms is the control - factor of the lithiation process in Si electrodes and sugge st that the lithiation rate can be enhanced by introducing vacancies in Si. Huang and co - workers performed first principle calculations to highlight the role of vacancies at the beginning of the lithiation process of c - Si. By calculating the binding energy , they concluded that vacancies can enhance Li binding energy significantly and make the initial Li insertion process thermodynamically more favorable. The vacancy - assisted diffusion of Li during the initial stage of lithiation provides the thermodynamic p roof for the effect of vacancies , which is consistent with our MD simulation results. In addition, the MD results also show more dynamics information . We also calculated the time required to fully lithiate c - Si structures with 5 % Si vacancies, shown in Ta ble 3 - 1 . As expected, the lithiation rate becomes faster in all cases compared to the structures without vacancies. The anisotropic effect also becomes less pronounced as the vacancies disrupt the ordered crystalline network. Figure 3 - 4 (a) Movement of reaction fronts with 2 %, 4 %, 6 %, 8 %, and 10 % Si vacancies with respect to time (b) Reaction rates with corresponding Si vacancy concentrations. All information w as obtained during lithiation of c - Si wit h (100) orientation at corresponding Si vacancy concentration at 1500K 44 T o elucidate the impact of Si vacancy concentration on the lithiation rate, we performed MD simulations of c - Si with 2 %, 4 %, 6 %, 8 %, and 10 % vacancies at 1500 K. The movement of th e reaction front and the radial distribution functions were analyzed to characterize the effect of Si vacancy concentration on lithiation dynamics. Fig . 3 - 4 (a) demonstrates the movement of the reaction front as a function of time at differ ent Si vacancy concentrations. As clearly indicated in the plot, the movement of the reaction front is faster with higher vacancy concentrations. With a linear fitting to describe the reaction front controlled lithiation dynamics, 44,47 the lithiation rate, , was obtained for each different vacancy concentration. Fig . 3 - 4 (b) shows that the lithiation rate exhibits an exponential relationship with respect to the vacancy concentration. This relationship can be explained by assuming the activation energy for the lithiation reaction, , is proportional to the energy required to break Si - Si covalent bonds due to vacancy generation, as: (3 - 3) where is the vacancy concentration, is the number of Si atoms, is the vacancy formation energy, is the Boltzmann constant, and T is the temperature. According to E qn. 3 - 3 , higher Si vacancy concentrations will result in faster reaction rates with an exponential relationship, which supports our data regarding th e movement of the reaction front and further confirms that the bond breaking of Si is the rate - limiting factor during the lithiation . To characterize the effect of Si vacancy concentrations on the crystal - to - amorphous phase transformation upon lithiation, structural evolutions during lithiation were investigated by comparing the of the snapshots at 1 ps and 1 ns for systems with 2 % and 8 % Si vacancies 45 in Fig. 3 - 5 . The initially sharp peak associated with the second nearest neighbors in diminishes and becomes broader with time, demonstrating an amorphization process. Figure 3 - 5 R adial distribution functions g Si - Si (r) at 1500K for c - Si with (100) orientation with 2 % and 8 % Si vacancies at (a) 1 ps and (b) 1 ns The difference of peak intensities between c - Si structures with 2 % and 8 % Si vacancies at 1 ps clearly indicates that the crystal - to - amorphous phase transformation occurs much faster with more Si vacancies. After 1 n s, the second nearest neighbor peak disappears and remaining peaks become broader for both cases, suggesting a complete transformation from c - Si to a - LiSi. To take advantage of vacancy accelerated lithiation dynamics, vacancies can be introduced into crys talline or amorphous Si experimentally via irradiation, chemical vapor deposition, and ion bombardment. 87 90 Previous expe rimental results and theoretical analysis also suggested the 46 vacancies and amorphization in Si induced by ion bombardment will lead to the formation of voids after high temperature annealing. 91,92 Amorphous Si with 10 % vacancies can transform into porous Si or relax to Si with less excess volume. 87 Thus, there must be a limit on the conc entration of vacancies which can exist in a Si anode that experiences electrochemical lithiation and delithiation cycles, warranting further research. 3.4.4 Random Diffusion of Li in Si is Concentration Dependent Experimental observations showed th at lithiation dynamics depend on various factors; both two - phase formation during lithiation and asymmetric Li diffusion rates suggest that it is possible that Li diffusivity is higher in Li - rich phase and lower in Li - deficient phases. 46,93 To further quantify the dependency of the Li diffusivity on Li concentrations, we computed the Li diffusivity at different L i concentrations by first defining the period where Li diffusion was characterized by 3.4.4. ( a ) Local Concentration Evolution In this study, the lithiation process in both a - Si and c - Si can be characterized as two stages: an initial mixing period, followed by a subsequent diffusion period. The initial mixing period is the time period when the covalent Si - Si bonding network breaki ng and Si volume swelling govern the lithiation dynamics. We defined the time of the initial mixing period to be from the beginning of the simulation until the local concentration become steady with a uniform concentration of 0.5. This time is summarized i n Table 3 - 1 . The initial mixing period is followed by a subsequent diffusion period, which is characterized by Li randomly diffusing in the a - Li x Si formed with a uniform Li concentration. We defined the subsequent diffusion period to be from the time when 47 the local concentration at different positions became uniform (end of the initial mixing period) until the end of our simulation time (~ 1 ns). The long time required to lithiate c - Si allowed us to analyze the local concentration evolution during lithiati on in more details. We investigated the concentration distribution with respect to the time and position during lithiation of c - Si with (100) surface orientations at 1 2 00 K and 1500 K as shown in Fig. 3 - 6 . Figure 3 - 6 Local concentration distributions with respect to (a) position at different time and (b) time at different positions at 1500 K. Concentration distributions respect to (c) position at different time and (d) time at different positions at 1200 K. All concentration profile w as obtained during lithiation of c - Si with (100) orientation at corresponding temperature Fig. 3 - 6 (a) shows the concentration distribution at different simulation times at 1500 K. A uniform Li distribution has been reached after 740 ps. Fig. 3 - 6 (b) shows the concentration evolution in several representative locations . Bin #1 is located in the initial Li r egion, bin #5 is 48 located at the initial Li/Si interface region, bin #9 is in the Si region close to the Li/Si interface, and bin #11 is located at the center of Si slab, which should be lithiated the last. As lithiation proceeds, the Li concentration decre ases in bin #1 and #5, while bin #9 starts to be lithiated after 25 ps and bin #11 starts to be lithiated after 7 0 ps. C oncentrations in all bins evolved toward 0.5, a1:1 ratio of Li and Si atoms, and reached 0.5 at 740 ps. After 74 0 ps, the concentration change in each bin becomes negligible, confirming a uniform concentration of 0.5. This process occurs similarly at 1200 K, reaching a concentration of 0.5 at 800 ps, as shown in Fig . 3 - 6 (c) and (d) . However, the lithiation at 1200 K shows that a slight co ncentration gradient concentration gradient still exists after 800 ps and becomes steady within our simulation time (~1 ns). T his non - uniform Li concentration may indicates that lithiation is still in progress at 1200 K after 1 ns. Theoretically, a longer simulation time will result in a uniform concentration of 0.5 at all positions. In our case, such complete mixing did not occur within 1 ns. Hwang et al . 78 observed a stress effect on the reaction front, more specifi cally, compression slows down the lithiation and eventually becomes stagnated. However, since our system becomes completely lithiated at 1500 K, it is more likely due to a combination of temperature and simulation time effects rather than the stress effect . Since Li diffusion in the initial mixing period deviates from random diffusion, it is unreasonable to compute the diffusion coefficient from averaged MSD calculations. Ho w ever, we noticed that in Fig . 3 - 6 (c) and (d) , the local concentrations during lith iation of c - Si with surface orientation (100) stopped changing after 8 00 ps. Locally, the structures have become amorphous with a uniform concentration from 800 ps to 1 ns. This time range with constant concentrations may be reliable for diffusivity calculations, since random walking governs the diffusion during this time inter val. Therefore, we computed the LMSD values at different bins using Eqn. 3 - 2 and calculated local diffusivities of Li using the Einstein relation in Eq n. 3 - 1 . The red dots in Fig. 3 - 7 49 (a) represent the Li diffusivities at different concentrations obtained from LMSD calculations . As local Li concentration increases, local Li diffusivity also increases. 3.4.4. ( b ) Random Diffusion of Li in Si is Concentration - Dependent In order to obtain a concrete trend between the concentration and diffusion coefficient over possible Li x Si structures, we also constructed various bulk structures of a - Li x Si with x ranging from 0.1 to 4.4 and performed NVT dynamics at high temperatures ranging from 900 K to 1500 K. Once the concentration became uniformly distribu ted, the averaged MSD calculations were performed to calculate the diffusion coefficient at corresponding concentrations, according to Eqn. 3 - 1 . Figure 3 - 7 (a) Diffusion coefficients of Li at different concentrations in logarithmic scale at 1200 K. Diffusion coefficients with corresponding concentration upon lithiation from bulk a - Li x Si is represented by blue dots. Red dots represent data obtained from local MSD calculations from initially c - Si with (100 ) surface orientation (b) Comparison Li diffusivity at with first - principle calculation and experiment Diffusion coefficients computed at 1200 K at each corresponding concentration from bulk a - Li x Si are represented by the blue dots in Fig. 3 - 7 (a) . The ove rall trend clearly indicates that an increase in concentration results in the increase in Li diffusivity. The local diffusivity data points 50 slightly deviate from bulk calculations due to the lack of statistics from our previous LMSD calculations. Interesti ngly, the relationship between the concentration and diffusion coefficient can be characterized by two distinctive regions, in which dramatic increase in Li diffusivity occurs below Li concentration of 0.8; and the increase becomes slower above Li concentr ation of 0.8. The overall trend indicates that increasing Li concentration results in faster Li diffusion, which is opposite from the intercalation type electrodes. 94,95 This phenomenon is likely due to the fact that semiconductor Si starts to show metallic behavior after the Li/Si ratio becomes larger than 1 (LiSi). The concept of concentration dependent Li diffusion in Si was recently revealed by AIMD calculations by Wang and co - workers . 58 Even though their results have captured the increase of D Li with Li content, the information was limited since the values of D Li in Li x Si were only calculated at x =1.0, 1.71, 3.25 and 3.75. Their results showed a linear trend between D Li and these Li concentrations, x . 58 With larger system sizes, longer simulation time, and a much broader concentration range of a - Li x Si (0.1 < x < 4.4), our work shows a nonlinear relationship and provides a more complete picture regarding concentration dependent Li diffusion in Si. The relationship of a faster Li diffusion with i ncreasing Li concentration can be is the underlying reason for exper imentally observed two - phase lithiation in both c - Si and a - Si. In situ TEM studies have revealed that during lithiation an atomically sharp (~1 nm) phase boundary forms, which abruptly separates the Li - rich region and Li - poor region. 43 To mimic this two - phase phenomenon, finite element models with concentration dependent diffusion coefficient have to be implemented. 46,96 However, the origin of this relationship is still missing. Our simulation results predict a lower diffusivity in Li - poor regions and explain the origin of the two - phase lithiation. This relationship may also be the underlying reason for lithiation/delithiation hysteresis. During battery cycling, the cut - off voltage is only determined by the surface concentration, while a 51 concertation gradient can still exist inside the e lectrode particle. During the lithiation stage, the initial diffusivity at low concentrations is significantly slower whereas the initial diffusivity of Li during the delithiation stage is faster. This difference will result in two different concentration profiles (during lithiation and delithiation) at the same surface concentrations, leading to different charge and discharge capacities at the same voltage. 3.4.4. ( c ) Room - Temperature Li and Si Diffusivity Calculation We have computed the diffusivity for three Li x Si concentrations, where x =0.1, 1 , and 4.4 a t temperatures from 900 K to 1500 K. From the diffusivities calculated at different temperatures for a - Li x Si, we extrapolated the diffusivities at room temperature at corresponding concen trations. The extrapolation is done by applying the Arrhenius expression for diffusivity as described in Eqn . 3 - 4 : (3 - 4) where E a is the activation energy , is the boltzmann constant, and T is the temperature. Li diffusivity at room temperature and at x values of 0.1, 1, and 4.4 are 2.002 10 - 16 , 1.965 10 - 9 , and 6.428 10 - 8 cm 2 /s , respectively , as shown in Fig. 3 - 7 (b) . Our results agree well with the result from AIMD calculations which provided the Li diffusivity at room temperature with x =1 to be ranging from 10 - 10 ~ 10 - 8 cm 2 /s. 56,58,97 Within the Li content range of x =1.00 to 3.75, the calculated D Li values using AIMD were in the range of 2.08 to 2.37 /s at 298K so the ReaxFF MD results and AIMD results are in good agreement . 58 Si diffusivity for the x values of 0.1, 1, and 4.4 are 2.270 10 - 20 , 1.277 10 - 11 , and 6.428 10 - 10 cm 2 /s , respectively. The experimental results for the diffusivity of Li in Si suggests the value of D Li to be from 10 - 14 ~ 10 - 8 52 cm 2 /s, 79,93,98 which agrees well with ReaxFF calculated values. Our result reveals that Si is almost immobile thus, the movement of Si atoms could significantly affect the lithiation rate. 3.5 Conclusions In summary, we have performed ReaxFF - based MD simulations to study the diffusion dynamics of both c - Si and a - Si. We discovered the lithiaton dynamics can be characterized as two stages; initial mixing period followed by subsequent random walk diffusion per iod. During the initial mixing period where crystalline - to - amorphous phase transformation occurs, Li diffuses faster in a - Si than in c - Si. During this period, Li diffusion in c - Si is governed by hopping diffusion between energetically favorable tetrahedral sites. The lithiation proceeds by layer - by - layer peeling off of (111) planes, which illustrates the location of (111) plane plays a critical role in lithiation - induced amorphization process. On evaluating the rate - limiting factor during Li mixing with Si, we found that introduction of Si vacancies resulted in accelerated lithiation rate whereas Li vacancies have negligible impact. The lithiation rate increases exponentially with the Si vacancy concentration, suggesting that Si - Si bond breaking is the rate - limiting factor, which is lowered by Si vacancies. Our results also revealed that Li diffusivity increases with concentration, which highlights the concentration dependent diffusion. This result provides the basis for the experimentally observed two - phase lithiation mechanism and is the underlying reason for lithiation / delithiation hysteresis. During the lithiation stage, the initial diffusivity of Li at low concentration is significantly slower than the initial diffusivity of Li during delithiation stage ( high Li concentration). Since the lithiation/delithiaiton capacity is directly related to the diffusivity and its dependence on Li concentration, the characteristic concentration dependent diffusion could significantly contribute to the lithiation/delithia tion hy steresis. These findings provide important 53 insight into understanding the dynamics upon lithiation in Si anodes, which could be utilized to design optimized batteries. 54 Chapter 4 Atomistic Simulation Derived Insight on the Irreversible Structural Changes of Si Electrode during Fast and Slow Delithiation 4.1 Summary Quantifying the irreversible chemical and structural changes of Si during cycling remains challenging. In this study, a continuous reactive molecular dynamics delithiat ion algorithm, with well - controlled potential gradient and delithiation rate, was developed and used to investigate the natural aluminum - oxide coated silicon thin - film. Fast delithiation led to the formation of dense Si netwo rk near the surface and nanoporosity inside the a - Li x Si, resulting in 141 % volume dilation and significant amount of Li trapped inside (a - Li 1.2 Si) at the end of delithiation process. In contrast, slow delithiation allowed the a - Li x Si to shrink by near - equ ilibrium condition, demonstrating no permanent inner pore with nearly Li - free structure (a - Li 0.2 Si) and minimal volume dilation (44 %). However, even without trapped Li, the delithiated a - Li x Si still exhibited higher volume (lower density) than the equilibrium structure with the same Li concentration, despite delithiation rate. The origin of this excess volume is the loss of directly bonded Si - Si pairs, which made the subsequent relithiation f aster. Based on the atomistic modeling and the quantified degradation mechanism, battery operating guidelines, including the delithiation rate and the depth of charge to avoid trapped Li and coating delamination, were suggested to improve the durability Si electrodes. This chapter is reproduced from the work published as: Kwang Jin Kim, James Wortman, Sung - Nano Letters 2017, 17 (7), pp 4330 4338. 55 4. 2 Introduction Delithiation is not a reverse process of lithiation. Regardless of the initial phase of Si being lithiated is crystalline or amorphous, fully lithiated silicide (crystalline or amorphous Li 1 5 Si 4 ) became amorphous upon delithiation. 33,35,36,38,99,100 It is obvious that the initial crystalline Si (c - Si) rarely went back to crystal after delithiation. 33,36,38,99 However, whether an amorphous Si (a - Si) structure returns to its original structure after lithiation and delithiation cycles remains unclear. Using in situ TEM, McDowell et al . 46 observed a clear phase boundary (indicating a two - phase lithiation process) during the first lithiation cycle of a - Si nanospheres, but no phase boundary during the second lithiation cycle. This indicates that the initial amorphous structure might be different from those after cycling. Most experimental techniques, including X - ray diffraction, transmission electron micros copy (TEM), and nuclear magnetic resonance (NMR), are not sufficiently sensitive to reveal the amorphous characteristics. Therefore, this paper used atomistic simulations with a newly developed delithiation algorithm, to illustrate the detailed structural differences after delithiation and the effect of these structural changes on the subsequent lithiation process. It quantifies, for the first time, the irreversible volume change, amount of trapped Li, generation and distribution of pores, and atomistic str uctural difference in the amorphous structures, upon delithiation. The irreversible volume change of Si nano - structures after cycling can be directly measured. Although nano - structures successfully mitigated fracture and pulverization of Si electrodes, 10,41 they sti ll do not return to their original volume after cycling. 46,101,102 For example, using in situ TEM, Ghassemi et al. 101 reported the diameter of the delithiated a - Si nanorods was 5 % larger compared to the initial diameter; and McDowell et al . 46 observed a 25 % volume dilation of delithiated a - Si nanospheres. The irreversible volume restoration becomes more interesting 56 when multiple lithiation/delithiation cycles are considered. Sun et al. 102 tracked the volume variations of Si - beade d - string structure for 18 cycles. In the first few cycles, the volume of the dilated Si - beads was 70 % larger than the original volume. Interestingly, the degree of dilation dramatically decreased to 10 % as more cycles were repeated. Two mechanisms, the formation of pores and voids inside the Si 103 105 and the residual Li trapped in Si, can both contribute to the overall irreversible volume dilation after cycling. Choi et al. 104 demonstrated that Si nanowires became porous and the pore size increased with subsequent cycle s. DeCaluwe et al. 105 observed the formation of nano - pores upon delithiation an d further proposed a pore collapse and regrowth mechanism upon multiple cycles. However, the volume of these pores/voids is difficult to quantify experimentally. The contribution from the trapped Li atoms to the volume dilation is even more difficult to qu antify. It was proposed that trapped Li atoms may be impossible to extract , since nano - structured Si electrode has higher density of defect sites, such as dangling bonds, which can bond Li more strongly. 106 109 Recently, Key and co - workers 35 confirmed the existence of trapped Li atoms after complete delithiation of Si electrode by analyzing the local structural changes with NMR, although the amount of Li is not yet determined. Besides contributing to the dilated volume, the trapped Li atoms are particularly interesting, as they directly lead to capacity loss. Furthermore, they may a lso significantly affect the final structure after delithiation and impact the subsequent lithiation process. Although the volume change induced solid electrolyte interphase (SEI) growth on Si electrode surface is also one of the main contributors to the i rreversible capacity loss, 24 26,44,45,110 it is out of the scope of the current study. This paper mainly focuses on the irreversible structural change of Si and reveal the detailed mechanical degradation mechanism. 57 Atomistic simulations in conjuncti on with experiments have played an important role in revealing the lithiation mechanism 24,32,33,34,37,38,41,43,58,63 ,66,95,96 but not the delithiation mechanism. It is relatively straightforward to simulate the lithiation process since simply putting Li and Si in contact will drive a spontaneous lithiation process due to the negative heat of mixing. However, it is challenging to accurately simulate the delithiation process since an external driving force is required to remove the Li from the lithiated silicon. M.K.Y. Chan et al. 51 developed a history - dependent Li insertion and removal algorithm based on energy minimization using density functional theory (DFT) , which limits the simulation size, so the long - range structural evolution and the information regarding volume contraction are difficult to obtain. Compared to DFT calculations, MD simulation with accurate interatomic potential, such as reactive force fie ld (ReaxFF), can simultaneously track the chemical, structural, and mechanical evolution for larger system size and longer time. Regarding delithiation simulation, Jung et al. 63 simulated the delithiation process by removing the Li atoms from the surface of a - Li x Si to generate a Li concentration gradient, which served as the driving force for delithiation. They revealed the formation of c - Si nuclei in the delithiated a - Li x Si matrix and demonstrated that the volume of delithiated a - Li x Si is larger than the origina l Si volume. However, removing all the Li from the natural in terms of volume contraction, structural evolution, Li diffusion in the Li - Si system. natur al ReaxFF - MD simulations was developed. The c ontinuous delithiation algorithm also captures the effect of different delithiation rate, which plays an important role in the irreversible st ructural change of delithiated Si. A lithiated a - Al 2 O 3 coating layer on Si was utilized as a reservoir to generate a driving force for Li to naturally diffuse out of the Li x Si. It is well known that a - Al 2 O 3 58 is a promising coating material with high thermal stability, high dielectric constant, and excellent ability to withstand the volume expansion. 113 Recently, the ReaxFF parameters for Li - Si - O - Al system were developed 74,114 an d used to simulate the lithiation process of coated - Si - nano - structures. 115 The mechanical failure of the coating during delithiation is also critical to Si electrode. With thi s new systematic delithiation algorithm, the fundamental reasons of degradation was investigated, by analyzing the relationship between the depth of discharge and corresponding volume and structural changes at different rates. Furthermore, the effect of ir reversible structural changes on subsequent lithiation processes was investigated. The Li trapping mechanism was also analyzed, which highlighted the important role of delithiation rates on irreversible capacity loss. 4.3 Simulation Methods 4.3.1 Reactive Force Field for Li - Si - Al - O - H System To perform delithiation simulation of fully lithiated a - Li 3.75 Si coated with aluminum oxide layer, we employed the reactive force field developed for Li - Si - Al - O - H system developed by Narayanan and co - wor kers . 74 Similar to the method Fan et al ., 59 implemented to develop force field for Li - Si system, Narayanan et al ., trained the parameters with DFT - calculated data for a wide variety of well - known condensed phases and clusters, as listed below ( description and references within Narayanan et al. 74 ) 1) Equations of state for pure Al (fcc, hcp, bcc, Al 2 O 3 ), surface energy of the fcc Al (111), charge distribution and dissociation energies of a number of Al - O - H clusters 2) Equations of state of Li (bcc, fcc, hcp, diamond, sc), LiH with sodium - chloride structure, di ssociation energies and charge distributions in Li 2 , LiH and LiH 2 clusters - Sn), SiO 2 - - cristobalite, stishovite), dissociation energies of single and double bonds of Si - Si and Si - O 59 in Si/O/H clusters, energies of various Si/O/H clusters as a function of valence angles Si - O - Si, O - Si - O, and Si - Si - Si and distortion energies of rings of Si/O/H clusters 4) Equations of state of Li - silicates: (a) Li 2 SiO 3 (orthorhombic), (b) stable Li 2 Si 2 O 5 (monoclinic) and (c) metastable Li 2 Si 3 O 5 (orthorhombic) - Li 2 O (cubic) and Li 2 O 3 (hexagonal) 6) Li aluminates: three polymorphs of LiAlO 2 7) Al silicates: three polymorp hs of Al 2 SiO 5 namely (a) andalusite (orthorhombic), (b) sillimanite (orthorhombic) and (c) kyanite (triclinic) For all the phases listed above (1) ~ (7), heats of formation as function of volume was calculated by the equation; (4 - 1) where is the total energy of a given volume V of the phase Li k Al l Si m O n subjected to particular strain . The energies of the constituent elements Li, Al, Si, and O (E Li , E Al , E Si , ) are those of the most stable phases at equilibrium calculated by DFT. All the parameters were trained against the heat of formation and was confirmed that structural properties and heats of formation for selected co ndensed phases agree well within the results of DFT calculations and with experimental results. Therefore, we employed the reactive force field developed for Li - Al - Si - O - H system to investigation the irreversible structural evolution upon delithiation of a - Li x Si with different rates. 60 4.3. 2 Molecular Dynamics Simulations of the Delithiation Process Figure 4 - 1 Model setup . (a) I nitial structure of fully lithiated a - Li 3.75 Si thin film sandwiched between fully lithiated aluminum oxide a - Li 8 Al 2 O 3 coatings, where spheres in yellow, green, red, and blue represent Si, Li, O, and Al atoms respectively (b) The computed open circuit voltage for a - Li x Si and a - Li x Al 2 O 3 . Continuously removing Li from the coating la yer outside of the buffer zone (white region) will generate the chemical potential driving force for the Li inside Si thin film naturally To study the volume variation, chemical reactions, and structural changes of a coated a - Li x Si thin film upon delithiation, a continuous delithiation algorithm was developed. All ReaxFF - based MD simulations were performed with the Large - scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). First, a slab structure of a - Li 3.75 Si was prepare d by melting (2500 K) and quenching (298 K) the c - Li 3.75 Si using NPT ensemble. Similarly, fully lithiated a - Al 2 O 3 (a - Li 8 Al 2 O 3 ) coating layers were obtained and attached to the both sides of a - Li 3.75 Si, mimicking a 61 coated Si - thin film as seen in Fig. 4 - 1 ( a ) . The obtained density of a - Li 3.75 Si and a - Li 8 Al 2 O 3 at 298 K is 1.12 g/cm 3 and 1.68 g/cm 3 , respectively. 99,115 Both values are in good agreement with experiments. The fully lithiated a - Al 2 O 3 (a - Li 8 Al 2 O 3 ) coating layer was utilized as a Li reservoir source, where Li atoms were continuously removed to drive delithiation normal to the Si thin film direction. More specifically, a buffer zone was defined as the region occupied by 10 % Al atoms in the coating la yer near the interface. Only the Li atoms outside the buffer zone were randomly natural delithiation responses, including atomic rearrangement and coating delamination, to be successfully captured without interrupting the a - Li x Si surface or the Li x Si/coating interface. At equilibrium, the chemical potential of Li should be the same in the coating and Si phases. This can be illustrated by the computed open circuit voltage (OCV) curve in Fig . 4 - 1 ( b ) , which is also the Li chemical potential change as a function of Li concentration. Decreasing Li concentration in the a - Li x Al 2 O 3 coating layer will decrease the Li chemical potential, causing a higher OCV in a - Li x Al 2 O 3 coating than a - Li x Si film. This Li chemical potential gradient will naturally drive Li atoms to diffuse out of a - Li x Si film in order to reach equilibrium. Based on this concept, a step - by - step delithiation algorithm was developed. At each delithiation step, a fix ed number of Li atoms, N = 100 in this work, was randomly removed from the lithiated coating layer outside of the buffer zone. Then, the structure was subject to NVT MD simulations for relaxation (with the Nose - Hoover thermostat and the velocity Verlet in tegration algorithm at a time step of 0.1 fs) at 900 K. 900K is about 50% of the melting temperature of Si (1687 K). Both experiments 80 and molecular dynamics simulations 56,58,62,78,112 (based on DFT or ReaxFF) have obtained diffusivity for Li - Si system at high temperature ranging from 600 K ~ 1500 K then extrapolated the room temperature diffusivity using Arrhenius equation. The extrapolated ro om 62 temperature Li diffusivity from the ReaxFF - MD simulation agreed well with experimental results, 79,98 which confirmed the usage of 900K in this study only accelerates the diffusion, thus reducing the simulation time without changing the diffusion mechanism. The relaxation time ( t) will allow the Li inside the a - Li x Si film to diffuse to the a - Li x Al 2 O 3 coating layer. To mimic the natural N/ t), t was varied. Delithiation steps were repeated until there is insufficient amount of Li atoms remained in the a - Li x Al 2 O 3 lay er to be removed, which is considered to be the fully delithiated state. Based on the geometry used in our simulation, a delithiation rate of 100Li/1 picosecond (ps) is equivalent to a current density, (4 - 2) which is much higher than the typical current densities used in experiments (10 - 3 ~ 10 - 6 A/cm 2 ). 105,116 The fast delithiation rate in MD simulation is expected and similar to the high strain rate used in MD simulations of deformation processes. 117 In order to determine t for the a - Li x Si structure to reach equilibrium at each delithiation step, we implemented a 1 D diffusion e quation for the slab structur e as described in the following equation (4 - 3) where C 0 is the initial concentration of a - Li 3.75 Si before delithiation and C s is the surface concentration after 100 Li atoms are removed from the a - Li 8 Al 2 O 3 coating. For D, constant Li = 1 - Initial Condition: C(x, 0) = C 0 for L < x < L Boundary Condition: C ( - L,t) = C(L,t) = C s at all t In this calculation, C 0 = 1.095×10 5 mol/m 3 , C 0 = 1.032×10 5 mol/m 3 , D = 5.15×10 6 cm 2 /s , L = 24 Å 63 diffusivity (5.15 10 - 6 cm 2 /s, determined in Chapter 3, S ection 3.4.4. (b) for a - Li 3.75 Si at 900 K) 112 by removing ~ 100 Li atoms was small enough. Although the init ial slab is only ~2.4 nm thick, the time estimated to reach equilibrium by solvin g the diffusion equation is ~ 30 ns. In other words, it still requires ~3 0 ns for Li to diffuse through the Si slab and reach the new uniform concentration in response to the concentration gradient ( c = 3%) caused by removing ~ 100 Li atoms. Such simulation time seems to be too long. Fortunately, a statistically uniform concentration within the simulation cell can be reached faster. The local Li concentration was calculated via a binning method as des cribed in Chapter 3, Section 3 . 3 . 3 . The fully equilibrated a - Li 3.75 Si structure shows a fluctuation of c = 0.06 in each bin, where c is defined as c = x /(1+ x ). Based on this analysis, we can consider an equilibrium conditions or a uniform Li concentration is reached when the Li concentration difference between adjacent bins and between the maximum and minimum values were both less than c = 0.06. The time to re ach this feature of equilibrium structure is approximately 0.9 ps based on local Li concentration comparison. Comparing the structures from negligible. T - Li x rates. 4.3. 3 Analysis of Volume Contraction, Pore Evolution, and Li Trapping In in situ TEM experiments, the total volume of the lithiated Si was determined by measuring the volume enclosed by the outer surface. Analogous to this definition, the vo lume of the slab structure, can be determined by the two outer surfaces. 46,101 Thus, the outer surface was first defined by averaging the coordination corresponding to the delithiation direction of the 64 outer most 10 % of Si atoms. Li concentration of the a - Li x Si film was defined by the numbers of Li and Si atoms within the two outer boundaries, as x in Li x Si. The local concentrations were obtained by dividing the a - Li x Si film along the delithiation direction into 9 bins with equal size. Upon delithiation, a - Li x Si structur es can become porous and the coating layer can also delaminate. Therefore, it is necessary and insightful to separately calculate the pore volume, , and the volume truly occupied by a - Li x Si atoms, , as (4 - 4 ) to form a complete picture. is the volume of any empty space generated inside the a - Li x Si (pore) and the empty space between the a - Li x Al 2 O 3 coating layer and a - Li x Si, leading to coating delamination. is computed via the Connolly volume method 118,119 implemented in Material Studio. surface area of molecules in the field of chemistry and biology. 118,119 In the schematic, the accessible surface area is drawn with green dashed line, which is ge nerated by tracing the center of the probe (in green) as it rolls along the surface of the atoms. Similarly, in our calculation, the volume within the locus of the probe center was considered as the volume occupied by the system. By applying the probe radi us of 1.2 Å (half of the Si - Si bond length), the vacancy with single atom size was excluded ( Fig . 4 - 2 (b) ) but other empty spaces larger than two atoms was treated as the pore volume ( Fig . 4 - 2 (c) , regions in yellow). is computed as . It represents the volume of truly occupied a - Li x Si by interconnected Li and Si, within the distance of the first nearest neighbor bond distance. 65 Figure 4 - 2 Sc hematic representation of Connolly volume calculation with (a) p robe (b) structure with internal pore smaller than single atom size and (c) system with internal pore larger than single atom size (considered as isolated - inner pore, region in yellow) 4.3. 4 Comparison between a - Li x Si at Equilibrium and Relithiated a - Li x Si In addition to the slab model for delithiation, bulk a - Li x Si with x ranging from 0.1 to 3.75 were prepared by heating and quenching method using NPT ensemble to obtain the equilibrium a - Li x Si structures. To study the effect of the structural changes on the subsequent lithiation cycles, the simulation was restarted from t he delithiated a - Li x Si slab after the slow delithiation. The slab was sandwiched in between two Li slabs, a similar geometry used in our previous lithiation simulation, 105 to be fully re - lithiated to Li 3.75 Si ( c = 0.789) during the 400 ps long NVT MD simulations. 66 4.4 Results and Discussion 4.4.1 Delithiation Proceeds with Different Rates Figure 4 - 3 Atomic Structure Evolution. Structural snapshots at the specified delithiation steps during the continuous delithiation process with the (a ) - ( c) fast and (d ) - ( f) slow ra tes. In the figure, spheres in yellow, green, red, and blue represent Si, Li, O, and Al atoms, respectively 67 The newly developed continuous delithiation algorithm was used to track the atomistic structure evolution of a coated - Si - thin film subject to fast ( shown in Fig . 4 - 3 ( a ) - ( c ) ) and slow (shown in Fig . 4 - 3 ( d ) - ( f ) ) delithiation rates. At time t = 0, both the a - Li 3.75 Si film and a - Li 8 Al 2 O 3 coating layers were fully lithiated ( Fig . 4 - 1 ( a ) ). During each delithiation step, Li diffused from the a - Li x Si film to the a - Li x Al 2 O 3 coating layers in response to the chemical potential gradient induced by Li - removal from the coating layer, causing the a - Li x Al 2 O 3 coating layers to be replenished. As Li atoms continuously diffused out of the a - Li x Si, significant vo lume contraction occurred, accompanied by isolated inner - pores generation in the a - Li x Si and surface delamination. As shown in Fig . 4 - 3 , the initial a - Li 8 Al 2 O 3 coated a - Li 3.75 Si thin film responded differently to delithiation rates. Although the same amount of Li was removed from both structures, the volume of the delithiated a - Li 3.75 Si thin film was always larger in the fast rate. After 55 fast delithiation steps ( Fig. 4 - 3 ( c ) ), only negligible amount of Li remained in the a - Li x Al 2 O 3 coating layers, s o the delithiation simulation was terminated. In this completely delithiated structure, significant amount of Li still remained in the a - Li x Si film. In contrast to the delithiation with fast rate, the increased relaxation time (10 times slower) is sufficie nt for Li to diffuse out from the a - Li x Si to the a - Li x Al 2 O 3 layer. As a result, as shown in Fig . 4 - 3 ( d ) - ( f ) , replenishment of Li in the delithiated a - Li x Al 2 O 3 occurred, thus extending the delithiation steps up to 65. This led to a higher total delithiat ion capacity in the slow rate. Fig . 4 - 3 ( f ) shows the fully delithiated structure with the slow rate. Compared to Fig . 4 - 3 ( c ) , only negligible amount of Li remained in the a - Li x Si and the higher degree of volume contraction was observed. 68 4.4.2 Porous Structures Evolution and Coating Delamination Figure 4 - 4 Pore Structure Evolution. Structural snapshots of the pore structure and distribution in the highlighted a - Li x Si region during continuous del ithiation process with the (a) fast and (b) slow rate Pore structure evolution during delithiation can highlight the rate dependent irreversible structural change. Fig . 4 - 4 visualizes the pores formed in the a - Li x Si during delithiation at the given steps. During the fast delithiation, no indication of isolated inner - pores was observed until the delithiation step of 33, corresponding to a - Li 2.8 Si. Then, isolated inner - pores formed inside the a - Li x Si and near the interface between a - Li x Si and the coating laye r, as shown in the structure after 40 delithiation steps. As the delithiation proceeded, the isolated inner - pores and interface - pores continuously collapsed and reformed but gradually increased in size. After 50 delithiation steps, in a - Li 1.5 Si, the pore s ize became significantly larger and the interface - pore size grew faster. 69 Eventually, most of the isolated inner - pores agglomerated with the interface - pores, generating a huge delamination pore between the coating layer and a - Li x Si - film. In comparison, duri ng slow delithiation, the formation of isolated inner - pores inside the delithiated a - Li x Si was first observed after 35 delithiation steps in a - Li 1.8 Si, but the number of pores was much less than that found in fast delithiation. As delithiation proceeded, i solated inner - pores temporarily formed and disappeared, since Li atoms had sufficient time to diffuse and the a - Li x Si film can reach a fully relaxed state. At 45 delithiation steps with a - Li 1.0 Si, the formation of interface - pores was observed, but no isola ted inner - pores were left. Then, the interface - pore size rapidly increased around a - Li 0.4 Si due to the severe surface delamination. The fast and slow rates in the simulation represents equilibrium and non - equilibrium delithiaion process. Under slow rate, pores might be temporarily generated, but Li and Si atoms inside the a - Li x Si have sufficient time to diffuse and fill the pore. The continuous formation and collapse of pores indicate that the delithiation proceeds by the formation of thermodynamically sta ble a - Li x Si. During the fast rate delithiation, Li and Si atoms have insufficient time to diffuse out, especially Si will only rearrange near small pores to minimize the number of dangling bonds. Continuous removal of Li leads to pit formation and porosity ; and eventually, the pore rapidly grows and agglomerates into larger voids. This confirms the nano - pores generation is a non - equilibrium process. The evolution of porosity in Si upon delithiation resembles the selective electrochemical de - alloying process, where the evolution of nanoporous structure depends on alloy composition, particle size, and de - alloying rate. 103,120 Chen et al . investigated the spontaneous nanostructures evolution in de - alloyed Li - Sn system and demonstrate that increasing de - alloying rate results in 70 nanoporous structure and pores. 120 The relationship between the pore size and diff usivity of the atoms upon de - alloying can be expressed as (4 - 5 ) where the represents the characteristic length scale of the nanostructure, D s is the diffusivity of adatom (Si in this case), and F is the stripping rate (delithiation rate). 103,121 Thus, higher delithiation rate and slow diffusivity lead to smaller nanostructure features a nd more porosity while slow delithation rate and faster diffusivity lead to larger structure feature and less porosity. Even though further quantitative analysis is required, our simulation clearly captures and agrees well with this scaling law, which can also be used to control Si morphology experimentally. The interface delamination began to grow at the stage of charge (SOC) of a - Li 2.2 Si and a - Li 1.0 Si, under fast and slow delithiation rate, respectively, suggesting the a - Al 2 O 3 coating delamination occurs earlier with faster delithiation rate. The failure mechanism of the a - Li x Al 2 O 3 coating layer depends on the initial geometry and the mechanical constraints during lithiation/delithiation. During lithiation, the a - Al 2 O 3 coating on Si - nano - wires can crack, b elow a critical thickness, due to the volume expansion of Si. 115 During delithiation, it could delaminate, buckle, or crack. However, the slab model with 1 - D delithiation dire ction implemented in this study is analogous to a Si - thin film subject to thickness change during cycling (no in - plane deformation). Although this geometry is likely to prevent buckling or cracking, it is interesting that the interface delamination occurs at the a - Li x Si/a - Al 2 O 3 interface. This is different from the a - Li x Si/a - C coating, where the delamination occurs inside a - Li x Si phase rather than at the interface. 122 71 4.4.3 Quantifying the Volume Contribution Experimental evidence as well as MD simulations clearly indicated that the total volume of the starting material is not restored after delithiation. 46,63,101 However, contraction analysis analogous to experimentally measurements imposes significant limitation on understanding the effect of delithiation rates and corresponding internal volume a nd structural changes buried inside. Therefore, we extended our study to separate and and quantified their contribution to the of the delithiated a - Li x Si, by the methods introduced in Section 4.3 . 3 . Figure 4 - 5 Volume and Concentration Evolution. (a) The total volume (V total ) and truly occupied a - Li x Si volume (V true ) change normalized with respect to the unlithiated Si and (b) Li concentration and the pore volu me contribution (V pore / V total ), upon delithiation with the fast and slow rate 72 Fig . 4 - 5 ( a ) plotted the and dilation percentage with respect to the initial un - lithiated Si during delithiation at different rate. The overall contraction trend resembles . The difference between and stems from , the total volume of all the inner - pores and interface - pores formed during delithiation. Fig . 4 - 5 ( b ) quantifies the evolution of to ratio as well as the Li amount left in the a - Li x Si film. During the initial stages of delithiation with fast rate (solid triangles on Fig . 4 - 5 ( a ) ), remained relatively constant and began to decrease after 20 delithiati on steps, at a - L 3.2 Si. The volume contraction continued throughout the remaining delithiation steps and at the final stage, the volume of the delithiated a - Li x Si became 141 % dilated with x = 1.2 residual Li left inside. The change of volume during delithiation with slow rate (solid dots in Fig . 4 - 5 (a ) ) is characterized by three stages. Initially, remained relatively constant until a - Li 3.2 Si, then the volume began to decrease at a constant rate. At approximately 50 steps, the degree of contraction decreased and leveled out around 44 % dilated with x = 0.2 residual Li left in the a - Li x Si core. The simulation not only captured experimentally observed trapped Li and po re generation but also provided additional quantified information and internal structural change details. Apparently, faster delithiation rate leads to more Li trapped inside the a - Li x Si core. Although the trapped Li inside a - Li x Si core is minimum after sl ow delithiation, the total volume still did not return to the original Si volume and ~ 44 % volume dilation was observed. However, the contribution of this volume dilation was not just from the porosity, which was only ~10 % of the . Even showed 17 % dilation after full delithation at slow rate. This suggests the fully delithiated a - Si may be different from the initial unlithiated a - Si. 73 4.4.4 Dilated a - Li x Si Exhibits Faster Lithiation Rate in the Second Cycle Figure 4 - 6 Difference in Amorphous Structures. (a) a - Li x Si structures formed during delithiation have lower density than the equilibrium structures. Density corresponding to the delithiated a - Li x Si upon delithiatio n with fast and slow rate are represented by blue and red dots. Green dots represent density obtained from the equilibrium a - Li x Si structures (b) RDF of g Si - Si (r) at 900 K of delithiated a - Li x Si upon delithiation with fast and slow rate and equilibrium a - Li x Si, all at concentration 2.8. Local concentration distributions at different stages of lithiation of (c) delithiated a - Si with slow rate and (d) equilibrium a - Si at 1200 K With quantified and Li composition in the a - Li x Si core, the density of the delithiated a - Li x Si structures with fast and slow rates are computed and compared with that of the equilibrium structure. Fig . 4 - 6 ( a ) clearly indicates that delithiated a - Li x Si with both fast and slow rates have lower density compared to the equilibr ium a - Li x Si structure at the same Li concentration. To 74 further investigate the structural differences, the radial distribution function of Si - Si, Li - Si, and Li - Li were analyzed for a - Li 2.8 Si structures, which had no indication of isolated inner - pores or su rface delamination. g(r) for Li - Si and Li - Li pairs were very similar with the equilibrium structure. However, the Si - Si distance, g Si - Si (r), of the delithiated a - Li 2.8 Si with fast and slow rates were different from that of the equilibrium a - Li x Si structur e, as shown in Fig . 4 - 6 ( b ) . The difference between the intensity of the peaks corresponding to the first and second nearest neighbor clearly indicates that there are less number of first and second nearest Si - Si bonds in the delithiated a - Li 2.8 Si structur es with both rates. Therefore, the excess volume in the delithiated a - Li x Si is due to the loss of directly bonded Si - Si pairs upon delithiation. The loss of directly bonded Si - Si pairs could impact the re - lithiation process. Kim and Qi demonstrated that b reaking of the Si - Si bond is the rate limiting step of lithiation and increasing Si vacancy is one of the methods to create broken Si - Si bond in order to accelerate Si lithiation. 112 To reveal the effect of the structural changes on subsequent lithiation rate, the a - Li x Si structure obtained after slow rate was re - lithiated ( Fig . 4 - 6 ( c ) ) and compared with the first lithiation of an equilibrium a - Si with the same number of atoms ( Fig. 4 - 6 (d) ). As shown in Fig . 4 - 6 ( c ) - ( d ) , delithiated a - Li 0.2 Si is completely mixed after 190 ps, whereas 300 ps is required to lithiated the equilibrium a - Si to become fully mixed in the first cycle. This clearly indicates that the loss of directly bonded Si - Si pairs and maybe the small amount of residual Li jointly make the delithiated a - Li 0.2 Si exhibit faster lithiation rate in the subsequent cycle. This is likely to be the underlying reason for the change from two - phase lithiation process 46,47 in the first cycle to one - phase lithiation process in the subsequent cycles. 46 75 4.4.5 Si Cage as the Origin of Li Trapping - Induced Irreversible Capacity Loss Structural configurations shown in Fi g. 4 - 3 and Li concentration evolution of the a - Li x Si shown in Fig . 4 - 5 ( b ) clearly indicate that different amount of Li remained in the delithaited a - Li x Si depending on the delithiation rates. In the case of the fast delithiation, even upon complete delithiation, there was a significant amount of Li remaining in the a - Li 1.2 Si whereas negligible amount of Li remained in the a - Li 0.2 Si at the end of slow delithiation process. Figure 4 - 7 Si Cage and Trapped Li. (a) Local Li concentration distribution at the final stage of delithiation with the fast (55 step) and slow (65 step) rate (b) Li motion tracking with various initial positions during fast delithiati on. In the figure, Si and Li atoms are excluded to improve the clarity. Spheres in green, red, and blue represent Li, O, Al atoms, respectively 76 During fast delithiation, the movement of the six individual Li atoms, at different depths inside the a - Li x Si th in film, was tracked. The trajectories shown in Fig . 4 - 7 ( b ) indicate that, Li # 1 ~ # 4 all diffused out of the a - Li x Si to the a - Li x Al 2 O 3 coating, and it took longer time for the Li deep inside the film to diffuse out. However, the Li atoms deeper inside the a - Li x Si film (Li # 5 and Li # 6) never diffused to the surface of the a - Li x Si, despite the driving force. As a result, the Li concentration near the surface was significantly lower than the core region. This non - uniform concentration was kept toward th e end of the fast delithiation process, as shown in Fig . 4 - 7 ( a ) for the a - Li 1.2 Si structure. The lower Li concentration on the surface in turn hinders Li diffusion, since Li diffusivity is lower at low Li concentrations. 112 Therefore, a cage - like locally dense Si network was formed near the surface, in which the low Li concentration prevented other Li atoms trapped inside to escape the caged surface, as demonstrated by the motion of Li # 5 and Li # 6. In comparison, durin g the slow delithiation, Li had sufficient time to diffuse from the a - Li x Si to the a - Li x Al 2 O 3 coating layer, preventing the formation of Li concentration gradient and allowed the delithiated structure to reach its fully relaxed state. Therefore, Li continu ously diffused out from the a - Li x Si and formed a - Li 0.2 Si with uniform Li concentration, as shown in Fig . 4 - 7 ( a ) . Since the number of available Li atoms is directly related to the capacity, slow delithiation will minimize the irreversible capacity due to the number of Li trapped in the a - Li x Si. Another efficient way to minimize the capacity loss is to avoid protective coating delamination by restricting the delithiation amount. For the slow delithiation rate (assuming equilibrium was reached in experiments ), the surface delamination did not occur before Li 1.0 Si. Therefore, if the coated Si - thin - film electrode can be cycled between a - Li 1.0 Si ~ a - Li 3.75 Si, coating delamination can be prevented. This can be achieved by controlling the upper cutoff voltage. Eve n though controlling the operating voltage limits the Si anode capacity, 9,33 it can beneficial in terms of the 77 overall battery life since it prevents the protective coating delamination from the severe volume contraction. Although this method will only use 73 % of the theoretical capacity provided by Si electrode, the capacity is still much higher than graphite electrode. 4.5 Conclusions In this study, a ReaxFF - based MD delithiation algorithm with controlled delithiation rate was developed. The delithiation response of aluminum - oxide coated Si thin - film was simulated under s low and fast rates, representing deliathion process near and far away from equilibrium. Characteristic structural features of the Li - Si system regarding the irreversible volume change, amount of trapped Li, generation and distribution of pores, and atomist ic structural difference in the amorphous structures, upon delithiation were investigated. During the fast delithiation, due to insufficient diffusion time, a cage - like Si - rich structure was formed near the surface. Since Li diffusion in Si - rich phase is s lower, the Si - cage trapped significant amount of Li inside the Si thin film. Therefore, at the end of delithiation, the composition is Li 1.2 Si and the volume dilation is ~141 %. In contrast, the slow delithiation gave Li sufficient time to diffuse out. Thu s, an almost Li - free structure, Li 0.2 Si, with 44 % volume dilation was obtained, and n o permanent inner pores, except coating delamination was observed. Even when there is no trapped Li, the delithiated amorphous Li x Si always exhibited larger volume (lower density) than the equilibrium structure with the same Li concentration, regardless of the delithiation rates. This is due to the loss of directly bonded Si - Si pairs, which made the delithiated a - Li 0.2 Si exhibit faster lithiation rate in the next cycle. Q uantifying the trapped Li clearly indicates that fast delithiation resulted in higher irreversible capacity loss compared to the slow delithiation due to the significant amount of Li 78 trapped in the a - Li x Si. This study also suggests that controlling the upp er cutoff voltage to keep the coated Si - thin - film electrode to be cycled between a - Li 1.0 Si ~ a - Li 3.75 Si will benefit the overall battery life since it prevents the protective coating delamination. 79 Chapter 5 Reactive Force Field Evaluation for Lithiation/Delithiation in Si - O 5.1 Introduction Silicon Monoxide (SiO) is another promising candidate for anode material with significantly higher capacity than graphite and enhanced cycle life compared to Si due to its unique microstructure and formation of irreversible sub - phases that can alleviate th e volume expansion upon lithiation/delithiation. 107 - 123 The microstructure of SiO has been debated over the past few decades with two main models. In the random b onding model (RB), SiO is considered as a single - phase material with randomly distributed Si - Si and Si - O bonds. 123,124 Random mixture model (RM) assumed SiO is composed of randomly distributed n - Si grains wi thin a SiO 2 matrix, implying a two - phase mixture. 125,126 Recently, experimental observation and theoretical calculations have suggested that solid SiO is thermodynamically unstable and undergoes a disproportionation reaction, where SiO segregates into Si and SiO 2 clusters surrounded by th e Si - suboxide matrix under heat treatment , suggesting the RM model. 129,131,137 To elucidate the controversial microstructure of SiO, Schulmeister et al . 130 performed structural analysis of a - SiO using TEM, EELS, and ESI and revealed that SiO is rather inhomogeneou s mixture of a - Si and a - SiO 2 than single - phase compounds (a RM model) . Furthermore, Park et al . 129 investigated the structural evolution of SiO upon heat treatment at high temperature ranging from 800 ºC ~ 1200 ºC and observed se gregation of thermodynamically unstable SiO into well - distributed nanocrystalline Si grains within an a - SiO x matrix (a RM model) , also known as the disproportionation reaction. 80 Figure 5 - 1 (a) EELS profiles (Si - K edge) taken from the three different regions in the HAADF - STEM image (b) Experimental (a ) - ( c) and simulated (d ) - ( f) ABED images corresponding to Si ,SiO 2 , and interface regions (Adapted from reference [ 133 ]. Copyright © Nature Communications 2016) Direct evidence on the atomic - scale disproportionation reaction and existence of the distinctive interface in SiO were proved by local structural analysis of SiO using angstrom - beam electron diffraction (ABED) and synchrotron high - energy XRD (HEXRD) . As shown in Fig. 5 - 1 (a) , EELS profiles (Si - K edge) taken from the dark (a - SiO 2 ), interface, and bright regions (a - Si) clearly indicates th at each regions have cha racteristic Si bonding environment. Furthermore, complicated bonding nature at the interface and ABED patterns were computationally investigated using Molecular Dynamics and Reverse Monte Carlo Method, 133 which confirmed the presence of suboxide - type tetrahedral coordinates (Si - (3Si,O), Si - (2Si,2O), and Si - (Si,3O)) in the interface regions. Elec trochemical reaction of SiO is characterized by the formation of irreversible sub - phases and reversible Li x Si. 29,127,131,135 Various experimental methods including NMR, 127,135 XPS, 127 electrochemical dilatometry were used to analyze the structure before and after the 81 electrochemical lithiathion/delithiation and verified the existence of Li - silicates, mainly Li 4 SiO 4 , and Li 2 O. Based on the initial capacity, irreve rsible capacity, and presence of irreversible Li - silicates and Li 2 O, lithiation of SiO was proposed to proceed by lithiation of SiO x sub - oxide with generation of various Li - silicates, followed by lithiation of c - Si or a - Si into Li x Si with continuous format ion of Li 4 SiO 4 and Li 2 O. Since the densities of Li 4 SiO 4 and Li 2 O are much higher than Li x Si, they are considered to act as a buffer to accommodate the large volume expansion. Jung and co - workers 13 further investigated the roles of various Li - silicates (Li 2 Si 2 O 5 , Li 6 Si 2 O 7 , and Li 4 SiO 4 ) and Li 2 O using DFT and validated oxygen atoms reduce the space occupied by Li, thus responsible for the alleviated volume expansion. Also, it was suggested that Li - silicates (mainly Li 4 SiO 4 ) are dominant over Li 2 O as irreversible sub - phases. Finally, Li diffusivity calculation using climbing - image nudged elastic band (CI - NEB) revealed Li diffusion in Li 2 O is at least 2 orders faster than that of other silicates, highlighting the importance of Li 2 O in mitigating the volume expansion and increas ing the rate performance. Evolution of the RM - two - phase microstructure upon lithiation and failure mechanism upon delithiation still remains unclear due to the limitation of experiment methods and computational power. Experiment methods are not sensitive enough to capture the atomic - level evolution of the dynamic structural changes upon lithiation/delithiation. In terms of simulation using DFT , it is even more challenging since the RM - two - phase geometry is difficult to describe with limited number of atoms . Therefore, it is necessary to use atomistic simulations to simultaneously track and correlate the microstructural evolution and failure mechanism of SiO upon lithiation/delithiation. Prior to simulating the lithiation and delithiation dynamics of SiO , it is necessary to evaluate whether the reactive force field parameters can successfully capture the phase stability of Li - Si - O system. Therefore, we evaluated the performance of reactive force field 74 82 developed for Li/Al/Si/O/H system (ReaxFF used in delithiation simulation from chapter 4) to check its app lication to simulate lithiation/delithiation of SiO. 5.2 Simulation Methods 5.2.1 First - principles DFT Calculations To generate the optimized RB - single - phase structures of a - Si, a - SiO, and a - SiO 2 using first - principle DFT calculations, AIMD was performed, as implemented in the Vienna Ab Initio Simulation Package (VASP). Projector - Augmented - Wave (PAW) potentials were used to calculate the interaction between ion cores and valence electrons. For exchange - correlation functional, generalized gra dient approximation (GGA) in the Perdew - Burke - Ernzerhof (PBE) was utilized. Due to the large simulation size and amorphous nature, k - point mesh of 1×1×1 was employed to save an enormous amount of computational time during the ab initio molecular dynamics s imulations. From the convergence test with criteria of 1 meV/atom, 550 eV used as the energy cutoff. The initial RB - single - phase structures for a - Si, a - SiO, and a - SiO 2 were generated by randomly distributing Si and O atoms into a simulation cell with appr opriate ratio at a density of 2.26, 2.21, and 2.17 g/cm 3 , which values were adopted from first - principle calculation of Si - O system. 139 Once the initial structures were prepared, amorphous Si, SiO, and SiO 2 were generated by melting - and - quenching process using AIMD , followed by relaxation. First, the initial structures were completely melted at 2500K for 5 ps, which temperature is suffic iently higher than the melting point of a - Si, a - SiO, and a - SiO 2 . Then, the structure was sequentially quenched to the room temperature at a rapid rate of 200 K per 1000 MD time steps , where each MD timestep corresponded to 1.0 fs. Once the temperature reac he d the room temperature, each system was 83 allowed to equilibrate for 5 ps, followed by relaxation with energy cutoff of 550 eV and atomic force tolerance of 0.02 eV/Å. The calculated density of a - Si, a - SiO, and a - SiO 2 were 2.28, 2.19, and 2.23g/cm 3 , which were good agreement with the experiments and theoretical studies. 13,29,139 5.2.2 ReaxFF - based Molecular Dynamics Calculations To generate the optimized structures of a - Si, a - SiO, and a - SiO 2 using ReaxFF , density of a - Si, a - SiO, and a - SiO 2 were calculated by using the initial RB - single - phase structure generated in Section 5.2.1 . Each structure was first melted at a high temperature (2500 K) and then quenched to room temperature via NPT ensemble, resulting in density of 2.44, 2.76, and 2.29 g/cm 3 for a - Si, a - SiO, and a - SiO 2 , respectively. The optimized structure generated for a - SiO is a RB - single - phase material with uniformly distributed network of Si and O. Figure 5 - 2 Schematic of (a) random bonding and (b) random mixture a - SiO To construct a RM - two - phase SiO structure observed in experiments, three representative random mixture SiO structures were generated. To generate these structures, the microstructure of the initial structure was simplified into a model with a - Si embedded inside the a - SiO 2 matrix. Based on the den sity obtained by ReaxFF - NPT MD simulations, three independent structures with 84 1, 4, 9 embedded Si regions were engineered inside a simulation cell of 50 Å ×50 Å ×20 Å ( Fig. 5 - 2 (b) ). To represent the experimentally observed random mixture microstructure with the size of the Si cluster regions around 1 ~ 5 nm 29,140 , radius of each structures were designed to be 15.572, 7.786, and 5.191 Å. Each system was composed of 1600 Si atoms and 1600 O atoms with total density of 2.34 g/cm 3 . To compare the phase stability between the RB - one - phase and the R M - two - phase microstructure s , two RB - one - phase SiO structures with different densit ies of 2.34 and 2.76 g/cm 3 were generated. Each density was chosen to represent the structure directly comparable to the simplified RM - two - phase SiO with a linear combination of the densities and RB - single - phase structure optimized by ReaxFF - NPT simulation. To prepare the RB - single - phase SiO, simulation cells with 1600 Si atoms and 1600 O atoms with density of 2.34 and 2.76 g/cm 3 were prepared. Then, ReaxFF - NVT simulations were performed at 4000K for 10 ps, followed by continuous run at 300K for 10 ps. The volume was kept constant throughout MD simulations to maintain the density. Detailed dimensions for simulation cells are listed in Table 5 - 1 . Struct ure Microstrucure # of Si Clusters Radius of Si Clusters (Å) Total Density (g/cm 3 ) Cell Length (Å) # of atoms x y z Si O a - Si Random Bonding - - 2.44 - - - - - a - SiO 2 Random Bonding - - 2.29 - - - - - a - SiO Random Bonding - - 2.76 42.304 50 20 1600 1600 - - 2.34 50.047 50 20 Random Mixture 1 15.572 2.34 50.000 50 20 4 7.786 2.34 50.000 50 20 9 5.191 2.34 50.000 50 20 Table 5 - 1 Summary of density, microstructure, and simulation size of a - Si, a - SiO, and a - SiO 2 85 5.3 Results and Discussions 5.3.1 Direct Energy Comparison of ReaxFF versus DFT Figure 5 - 3 Energy calculation of a - SiO 0.5 , a - SiO, and a - SiO 2 with equation of states In addition to the large systems, bulk structures of a - Si, a - SiO ( , and a - SiO 2 with much smaller size (tot al number of atoms being less than 100 atoms) were also generated using the density predicted by ReaxFF - NPT simulation, followed by ReaxFF - NVT simulation at 4000K and 300K. The energies of smaller structures were calculated by DFT and ReaxFF to evaluate th e performance of the ReaxFF parameters. The energy of the DFT - optimized a - Si, a - SiO, and a - SiO 2 along with equation of states using DFT are presented in Fig . 5 - 3 . The energy of the optimized a - Si, a - SiO, and a - SiO 2 structures generated using first - principle DFT calculations are - 5.17 eV, - 13.96 eV, and - 23.39 eV, 86 f for a - SiO, which can be expressed as (5 - 1) where , , and represents the total energy per formula unit of a - Si, a - SiO, and a - SiO 2 , respectively. Structure Generation Method Energy Calculation Method Energy per Formula Unit (eV) Formation Energy (eV) Phase Stability of Random Bonding SiO a - Si a - SiO a - SiO 2 AIMD DFT - 5.173 - 13.959 - 23.387 0.321 Unstable ReaxFF - 4.395 - 10.336 - 17.312 0.518 Unstable ReaxFF - MD DFT - 4.928 - 7.716 - 21.929 5.713 Unstable ReaxFF - 4.651 - 14.620 - 19.901 - 2.344 Stable Table 5 - 2 Summary of energy calculation of a - Si, a - SiO, and a - SiO 2 from optimized structures generated by first - principle DFT calculations and ReaxFF - MD simulations Figure 5 - 4 Formation energy of a - SiO with (a) DFT optimized and (b) ReaxFF - MD optimized structure. The red and blue data points represent DFT and ReaxFF - MD energy calculation, respectively 87 Table 5 - 2 summarizes the information regarding the energy of a - Si, a - SiO, and a - SiO 2 calculated from the DFT - optimized and ReaxFF - MD optimized structures. Fig. 5 - 4 com pa res the formation energy calculated using different calculation methods for the two structures g enerated by AIMD and ReaxFF - MD , respectively . The f ormation energy of the DFT - optimized RB - single - phase structure using DFT method is 0.321 eV. This indicates that RB - single - phase SiO is thermodynamically unstable and the microstructure should segregate in to two - phase. Also, the formation energy of the AIMD generated RB - single - phase a - SiO calculated by ReaxFF method is 0.518 eV, which indicates ReaxFF can successfully predict the energetics of the thermodynamically unstable a - SiO structure generated by AIMD calculation. In contrast to the energy calculation from the AIMD - optimized structure which both calculation method agrees with each other, the energy calculation of the ReaxFF - MD optimized structure using ReaxFF predi cts a negative formation energy wherea s DFT method predicts a positive formation energy, as shown in Fig. 5 - 4 (b) . In other words, even though ReaxFF - MD optimized structure is unstable with positive formation energy as predicted by DFT method, ReaxFF is predicting this structure to be stable w ith negative formation energy. This is why this erroneous structured can be generated by the ReaxFF - MD. 5.3.2 Non - Physical SiO Structure Predicted by ReaxFF - MD In this section, the phase stability of the SiO with RB - single phase and RM - two - pahse microstructures was further evaluated by ReaxFF method. 88 Figure 5 - 5 (a) Energy Comparison of random bonding and random mixture a - SiO with its unique microstructure. (a) - # represents the microstructure o f a - SiO corresponding to the energy. In the f ) of a - SiO s tructures As clearly indicated from the energy plot ( Fig. 5 - 5 (a) ) , the energy of RB - single - phase SiO is significantly lower than those of RM - two - phase SiO. In other words, ReaxFF - MD predicts the RB - single - phase a - SiO is more thermodynamically stable than the RM - two - phase a - SiO. It is interesting to note ReaxFF - NVT simulations at density of 2.33 g/cm 3 ( Fig. 5 - 5 (a) - 4 ) results in a structure with a huge hole at the left corner of the simulation box, indicating the RB - sinlge - phase SiO is extremely unstable at this density (strong Si - O bond predicted by ReaxFF) and returns to it optimized state at densi ty of 2.76 g/cm 3 . Also, the formation energy of each structures was computed using the energies obtained from ReaxFF - MD NPT simulations for a - Si and a - SiO 2 . Regardless of the microstructures, ReaxFF predicted negative formation energ ies for all the 89 structures. This is not consistent with many reported experimental observations and contradicts the DFT calculations, which predicted RB - single - two - phase is energetically unstable. 5.3.3 Reason for the Errors in the MD Results The discrepancy betw een the optimized structures and corresponding energetic using first - principle DFT and ReaxFF is due to the scope of the training set which ReaxFF is trained on. As described in Chapter 3, S ection 3.3.1, van duin et al . 73 trained the R eaxFF for Si and Si - Oxide system for a wide variety of well - known condensed phases and clusters of Si and SiO 2 . Figure 5 - 6 Energy calculation using DFT and ReaxFF methods for (a) three different Si phases and (b) various silicon dioxide phases at different densities for ReaxFF fitting (Adapted from reference [ 73 ]. Copyright © The Journal of Physical Chemistry A 2003) Fig. 5 - 6 shows one example of the evaluation of the ReaxFF trained for Si and SiO 2 with different crystal structures based on equation of states comparison. As clearly shown in the energetics, ReaxFF successfully captures the energy change upon different densities of different 90 2 phases ( coesite, and stishovite SiO 2 ). Even though this set of ReaxFF parameters can accurately describe the chemistry and physical environment of the Si and SiO 2 phases which were accurately fitted to against the quantum calculation, they perform poo rly in the region s which are not included in the training set , such as SiO . As explained, the RB - single - phase SiO is thermodynamically unstable phases. Since the set of ReaxFF parameters was not carefully fitted or learned from this unique environment of SiO, it is extremely difficult to successfull y capture the properties of SiO accurately. This is the reason behind the contradiction of the formation energy calculated from the ReaxFF - optimized structure. Therefore, it is necessary to refit the ReaxFF parameters with a range of metastable a - SiO x stru ctures between a - Si and a - SiO 2 to properly describe the phase stability of SiO. 5.3.4 Design of the Training Set for Li - Si - O System To successfully describe the phase stability and further simulate lithiation/delithiation of SiO, the training set wh ich describes the Li - Si - O system needs to be extended and include range of a - SiO x structures between a - Si and a - SiO 2 with corresponding energies calculated from first - principle DFT calculations. Parametrization of ReaxFF with extended training set should be proceeded via a two - step procedure. First, to capture the energetics of unique SiO x microstructure, it is important to successfully describe all possible environments between the composition range of x = 0 ~ 2 for SiO x . Also, structures within this range must be allowed to evolve to the optimized structures with the correct energies at the corresponding composition. Therefore, it is necessary to optimize Si/O parameters by fitting against equations of state of several representative metastable phases including a - Si, a - SiO 0.5 , a - SiO, a - SiO 1.5 , and a - SiO 2 . Once the Si/O parameters were tuned, then 91 Li/Si/O parameters also require modification to accurately simulate the lithiation/delithiation of SiO x system. For this purpose, Li/Si/O parameters should al so be refitted against equation of state of a - Li x SiO 0.5 , a - Li x SiO, a - Li x SiO 1.5, and a - Li x SiO 2 where x ranges from 0 to 6 . Similar to the method used in the original training set, 74 these equations of state should be computed for volume changes ranging from 50 % compression to 20 % expansion. 5.4 C onclusions In this study, the performance of ReaxFF developed for Li/Al/Si/O/H system was evaluated to check its transferability to describe the phase stability of SiO. Formation energy of DFT - optimized and ReaxFF - MD optimized SiO structure was compared u sing DFT and ReaxFF calculation method. Formation energy calculation of DFT - optimized SiO structure indicates that regardless of the calculation method, the structure is predicted to be thermodynamically unstable with positive formation energy. However, the sign of the formation energy of the ReaxFF - MD optimized SiO structure differs depending on the calculation method. DFT method predicts a positive formation energy whereas ReaxFF method predicts a negative formation energy, which discrepancies stems fro m the lack of training set describing the metastable a - SiO regions. Since the parameters describing the reaction of Si and O was only learned from equation of state of various c - Si and c - SiO 2 and some clusters, it is extremely difficult for ReaxFF to predict metastable a - SiO regions with completely different structural motifs than Si and SiO 2 . Therefore, to successfully describe the phase stability and further perform lithiation/delithiation of SiO, it is necessary to expand the training set with a ran ge of metastable a - SiO x including a - Si, a - SiO 0.5 , a - SiO, a - SiO 1.5 , and a - SiO 2 as well as their lithiated phases. 92 The training set for SiO x system which include different crystal structures of Si and SiO 2 as well as amorphous SiO 0.5 , SiO, and SiO 1.5 has b een generated. The new parameters will be generated via fitting procedure using genetic algorithm through a collaboration with Dr. Aktulga 93 Chapter 6 Generation of Li - Si Training Set for Machine - Learning Potential 6 .1 Introduction Up to this point, all the ReaxFF - based atomic simulations were performed using the ReaxFF developed for Li - Si 59 and Li - Si - Al - O - H system. 74 Although ReaxFF trained against DFT calculations can accurately describe the chemical reaction, methodology regarding the development of such force field has several disadvant ages. 141 143 The construction of accurate interatomic potentials (or f orce f ield) is a repetitive task of training against extensive first - principle even after an acceptable potential has been formed, extension to new systems is difficu lt because of the complex interdependence of all parameters which often requires introduction of new energy terms based on trial and error. To efficiently develop interatomic potentials which provide reliable energies and forces, currently a paradigm chang e is taking place in the development methodology. 143 Machine - Learning Potentials (MLP) 142 147 have been introduced as an alternative ap proach to obtain the accuracy of first - principle calculations and the efficiency of a force field based on functional forms. Different from the force fields which are developed on functional forms, MLP employs purely mathematical fitting techniques to cons truct a direct relation between the atomic configuration and corresponding energies based on the training - set consisted of accurate first - principle calc ulations. The advantage of MLP is that the development procedure can be well - automated so that extension of the potentials for new system can be performed with minimum human intervention. Also, the flexibility of the mathematical potential model allows the MLP to reach the accuracy that is comparable to the training set used in their construction. However, t he 94 characteristic mathematical potential model of MLP also causes a significant limitation. Due to the non - physical form of MLP, it is difficult to directly interpret the model parameters and describe the regions that are not included in the training set. Therefore, t he quality of the MLP significantly depends on the training set and it is most important to construct an extensive training set which samples all relevant atomic interactions and environments. In this chapter, systematic method to generate acc urate and extensive training set for Li - Si system is introduced. Li - Si system was specifically chosen because of its wide application in battery simulation since Si is widely studied as a promising anode material. Furthermore, lithiation of c - Si at room te mperature proceeds via solid - state amorphization, thus require investigation of amorphous systems. Generally, it is difficult to characterize the amorphous phases with experiment techniques since they are not sensitive enough to capture the wide variety of local structural motifs. First - principle calculations can provide insights regarding phase stability and atomic structures but due to the computational cost, the number of the atoms are limited to hundreds of atoms. Therefore, it is particularly important for amorphous phases to be studied in large scale with longer dynamics with first - principle accuracy and interatomic potential efficiency. In this purpose, extensive training set which MLP can be learned was developed and the performance of the training s et was evaluated by computing energies, voltages, chemical properties, and mechanical properties with DFT calculations. 6.2 Automated Construction of the Training Set with Crystalline and Amorphous Li x Si 6.2.1 c - Li x Si structures The crystalline phases with composition Si, Li, LiSi, Li 12 Si 7 , Li 7 Si 3 , Li 13 Si 4 , and c - Li 15 Si 4 were obtained from crystal structure database from Material Project. First - principle DFT 95 calculations were performed as implemented in the VASP with PAW method to des cribe the interaction between ion cores and valence electrons. For exchange - correlation functionals, GGA method was utilized. The energy cutoff was determined to be 550 e V. The optimized lattice parameters and K - point mesh are summarized in Table 6 - 1 . Pha se Space Group a (Å) b (Å) c (Å) k - point Mesh Si Fd m 10.861 10.861 10.861 4×4×4 LiSi I41/a 9.340 9.340 5.760 4×4×4 Li 12 Si 7 Pnma 8.554 19.657 14.310 4×4×4 Li 7 Si 3 C2/m 7.629 6.607 18.009 5×5×5 Li 13 Si 4 Pbam 7.972 15.143 4.450 6×6×6 Li 15 Si 4 3d 10.655 10.655 10.655 3×3×3 Li Im3 7.018 7.018 7.018 6×6×6 Table 6 - 1 Structural parameters for c - Li, c - Si, and c - Li x Si alloys at equilibrium 6.2. 2 a - Li x Si Structures To generate extensive training set which includes a wide range of local structural motifs of a - Li x Si system, I classified the training set into three categories based on their energy state s. These t hree categories are 1) Liquid - like a - Li x Si structures , 2) Near - Ground - State a - Li x Si structures , and 3) Intermediate a - Li x Si structures . The Liquid - like a - Li x Si structures represents the local structural motifs at the highest energy state . The near - ground - state a - Li x Si represents the equilibrium structures of metastable a - Li x Si with the lowest energy. The i ntermediate a - Li x Si structures represent the configurational space between the liquid - like and near - ground - state a - Li x Si . These three categories together cover the possible configurational space occurring in a - Li x Si system. Amorphous Li x Si structures at different energy states were generated using a combination of ReaxFF - based MD along with AIMD and optimization protocol in the framework of DFT . 96 Figure 6 - 1 Flow chart describing the autom ated protocol to construct the Li - Si training set Fig. 6 - 1 illustrates the flow chart of the automated protocol to construct the Li - Si training set. First, initial configurations of a - Li x Si structures with concentration ranging from x = 0.1 ~ 3.75 were prepared by approximating the densities with corresponding concentrations. T he densities of a - Si, a - LiSi, a - Li 12 Si 7 , a - Li 7 Si 3 , a - Li 13 Si 4 , and a - Li 15 Si 4 were calculated from melting - and - quenching process from their crystalline counterpart s. Similar to the process used in Chapter 3, S ection 3.3.3 , each structure was melted at high temperature of 2 500K, followed by quenching process to room temperature using NPT dynamics while the ratio of Li and Si was kept constant. 97 The calculated density of a - Si, a - LiSi, a - Li 12 Si 7 , a - Li 7 Si 3 , a - Li 13 Si 4 , and a - Li 15 Si 4 were 2.44, 1.91, 1.64, 1.48, 1.26, and 1.19 g/cm 3 , respectively. Figure 6 - 2 (a) Radial distribution function of c - Li 15 Si 4 and a - Li 15 Si 4 (b) Densities of a - Si, a - LiSi, a - Li 12 Si 7 , a - Li 7 Si 3 , a - Li 13 Si 4 , and a - Li 15 Si 4 calculated from ReaxFF - NPT simulations via melting - and - quenching process Linear fit ting of the density as a function of x was obtained based on the densities calculated for repre sentative concentrations and utilized to generate configurations at corresponding concentrations , as shown in Fig. 6 - 2 (b) . Each structure was generated with concentration interval of 0.1, resulting in 37 initial structures with concentration ranging from which sufficiently cover the entire concentration range. The initial configurations were generated using random generation method in material studio with the number of Si varying at different concentrations. Total 50, 30, and 20 number of Si were used for the concentration range x 1.0, 1.0 x 2.0, and 2.0 x 3.75 to ensure the total number of atoms were within ~ 150 atoms for efficient DFT calculations. Once the initial configurations of a - Li x Si at 37 concentrations were prepared, ReaxFF - MD simulations using NVT dynamics at high temperature (2500K) was performed for 10 ps to 98 completely melt the structure ( Fig. 6 - 1 Step 1 ) . Then, independent configurations were random ly chosen from the trajector y to generate liquid - like, near - ground - state, and intermediate a - Li x Si structures ( Fig. 6 - 1 Step 2 ) . Amorphous Li x Si structures were generated using AIMD and different optimization protocol . For AIMD , a 1×1×1 K - point mesh and energy cutoff of 450 eV were employed to save enormous amount of computational time. Besides the K - point mesh and energy cutoff for AIMD, all the DFT calculation setups were kept identical to those used in the calculation of crystalline phases. Liquid - like a - Li x Si structu res representing the highest energy state of the metastable a - Li x Si were generated by following Step 3 - a in Fig. 6 - 1 . Twenty independent configurations from the high temperature MD trajectories for each concentration were obtained and DFT single - point ener gy calculation was performed to correlate the local structural motifs to the corresponding energies. Total 740 liquid - like configurations with corresponding energies were generated and incorporated into the database . To obtain near - ground - state a - Li x Si structures which represen t the equilibrium state with the lowest energy, melting - and - quenching process were performed using AIMD ( Step 3 - b in Fig . 6 - 1 ) . Five independent configurations from the high temperature AI MD trajectories for each concentration were chosen and each configuration was quenched to the room temperature at rapid rate of 300 K per 10 00 MD time steps, where each MD timestep corresponded to 1.0 fs. Then the corresponding structures were relaxed with energy cutoff of 550 eV and atomic force tolerance of 0.02 eV/Å. To completely relax the structure and obtain near - ground - state structures, atomic coordinates, dimensions, and the shape of each structures w as allowed to relax. Total 185 near - ground - state configurations with corresponding energies were generated and incorporated into the database. 99 Finally, to sample intermediate a - Li x Si structures which covers the configurational space between the highest and lowest energy states, DFT relaxation with fixed volume were performed on the five independent structures obtained from the high temperature MD trajectories for each concentration ( Step 3 - c in Fig . 6 - 1 ) . All the configurations and corresponding energies during the relaxation , which represent the local structural changes from high energy to low energy states, were collected and incorporated into the database (total 87270 structures). These structures will be critical if both energy and force will be fitted through machine lear ning. 6.2. 3 Formation Energy and OCV for the Training Set The f ormation energy is an important parameter to judge the stability of a compound, which contribution be the total energy of a system calculated at T = 0 K. In case of Li - Si alloys, the formation energy is defined as (6 - 1) where x is the number of Li atoms per Si atom . is the formation energy per Si (formula unit of Li x Si) and is the total energy of the Li x Si structure per Si (formula unit of Li x Si). E Li and E Si are the energy of a single atom in the elemental body - centered cubic Li in elemental Si in a diamond lattice. 100 Figure 6 - 3 Formation energies of crystalline and amorphous Li x Si included in the training set. The training set is consisted of total 88201 structures The formation energies per Si atoms of the ground - state c - Li x Si and metastable a - Li x Si as a function of Li concentration are shown in Fig. 6 - 3 . As expected, thermodynamically stable ground - state c - Li x Si (green data points in Fig. 6 - 3 ) and liquid - like a - Li x Si (blue data points in Fig. 6 - 3 ) have the lowest and highest formation energies, respectively. The formation energies of near - ground - state a - Li x Si (purple data points in Fig. 6 - 3 ) are lowest among the a - Li x Si since they are the equilibrium structure for the metastable a - Li x Si. Also, they agree well with the formation energies calculated from amorphous structures generated by different protocol, 52,147 which confirmed the protocol to generate equilibrium a - Li x Si structure in this study successfully captures the local structural motifs and corresponding energies. 101 To further relate the energies to the experimental results, the OCV was computed using the following equation (6 - 2) x Si) is the formation energy obtained from DFT calculations at T = 0 K. Figure 6 - 4 Experimental 34,42 and calculated OCV vs. composition curves of Li/Si system at high temperature (lines in red) and room temperature (curves in blue) (Adapted from reference [ 34 ]. Copyright © Journal of the Electrochemical Society 2004) Fig. 6 - 4 represents the experimentally measured and computationally calculated OCV as a function of Li concentration ( x ) for crystalline Li x Si structures (red curve) and metastable near - ground - state a - Li x Si (blue curve). Dotted red curve represents the experimentally measured OCV at high temperature and solid red curve represents the computationally calcul ated OCV in this study. Similarly, dotted blue curve represents the experimentally measure OCV upon electrochemical lithiation of a - Si and solid blue curve represents the computationally calculated 102 OCV from our near - ground - state a - Li x Si structures. Both th e computed voltage profile agrees well with the experimentally measured voltages, which confirms that the 0 K voltage profile is a reasonable approximation and the near - ground - state a - Li x Si successfully represent the amorphous structures in equilibrium. Fin ally, the red data points in Fig. 6 - 3 represents the formation energies of the structure in between the high energy and near - ground - state a - Li x Si. For clarity, the figure only shows the lowest formation energies at all concentrations and the relaxation string for concentration 1 .0 and 2.5 . The area in yellow represents the configurational space which the intermediate a - Li x Si covers. From the plot, it is confirmed that our method covers all the possible configurational spaces (structural motifs) occurrin g at a - Li x Si . 6.3 Elastic Property Calculation to Validate Machine Learning Results In this section, the elastic properties were calculated for the near - ground - state a - Li x Si structures, which will not be included in the training set but will be used to evaluat e the accuracy of the machine learning potentials developed based on our training set provided in S ection 6.2 generate Li - Si potential through their machine learning techniques). The elastic properties of a - Li x Si are determined by computing the energetics of the deformed unit cells. Due to the isotropic nature of amorphous phases, different elastic constants can be deduced from the bulk modulus (B) and modulus C 11 . Therefore, bulk modulus (B) and modulu s C 11 To calculate the bulk modulus, uniform tensile/compressive stress ( , percentage) were applied on the a - Li x Si alloys to achieve volume variation. The relaxed energy of the deformed cells 103 were fitted to the equation E ( ) = E 0 + a 2 , where E 0 is the energy of the undeformed cell. Then, bulk modulus was calculated using formula; (6 - 3) wh ere V 0 is the volume of the unstrained cell and a is the coefficient of 2 in the quadratic fit. Similarly, modulus C 11 was calculated by imposing tensile/compressive stress ( , percentage) on one orthogonal axe and computing the energies of the deformed configurations. The relaxed energy of the deform cells were fitted to the fit to the expression E ( ) = E 0 + b 2 , modulus C 11 was calculated by using formula (6 - 4) Once the bulk modulus and modulus C 11 were then calculated using the expression; (6 - 5) (6 - 6) Fig . 6 - 5 (b) represents the bulk modulus of c - Li x Si and near - ground - state a - Li x Si structures at five representative concentrations ( a - LiSi, Li 2 Si, Li 3 Si, and Li 3.75 Si) along with the computed bulk modulus taken from the literature. 49,55 The bulk modulus of near - ground - state a - Li x Si structures decreases linearly, which successfully capture the elastic softening of Li - Si phases with increasing Li concentration . 55 104 Figure 6 - 5 Bulk modulus for a - Li x Si with comparison with literature (DFT) 49,55 Sim - Si, a - LiSi, a - Li 2 Si, a - Li 3 Si, and a - Li 3.75 Si were computed which agree well from the literature value. 49 Thus, the results further confirmed the training set generated in this work can successfully describe the elastic properties of Li - Si alloys. Such elastic information will be used to check the performance of the potentials developed using machine learning techniques, which is only fitted on the relationship between the structures and corresponding energetics. 6.4 Conclusions In this stud y, I systematically developed an extensive training set of Li - Si alloys, which includes thermodynamically stable crystalline and metastable amorphous phases. Our methodology is based on a combination of ReaxFF - MD and first - principle DFT calculation through a FireWork work flow. To represent a wide range of structural motifs, Li - Si data base with energy states. For the databased, 740 liquid - like a - Li x Si structu res at the highest energy and 185 105 near - ground - state a - Li x Si structures at equilibrium was prepared. Also, 87270 intermediate structures in between the two extreme energy states were collected. I demonstrated that our training set successfully describe the metastable a - Li x Si in different energy states which covers all the configurational space for Li - Si systems. The systematic method used herein is not limited to a specific material or amorphization mechanism and can be generally applied for simulating amorp hous materials, which can be further utilized to develop machine learning potentials which can simulate materials with first - principle accuracy and force field efficiency. - Si potential through their machine learning techniques. 106 Chapter 7 Conclusions In this thesis, atomic - scale MD simulations with Reactive f orce f ield were performed to obtain insights on the lithiation mechanism to accelerate the lithiation rate and elucidate the irreversible changes that are inherent at atomic - scale upon delithiation with rate effects. Also, the application of the reactive force field to the SiO x system was investigated, which allowed us to better understand and design an extensive training set fo r f orce f ield development. To elucidate the rate - limiting factor of lithiation, ReaxFF - based MD simulations to study the diffusion dynamics of both c - Si and a - Si were performed. The rate - limiting factor during lithiation of Si is the Si - Si bond breaking process and introducing Si vacancies exponentially increases the lithiation rate. Our results also revealed that Li diffusivity increases with concentration, which highlights the concentration dependent diffusion. Since the lithiation/delithiaiton capacity is directly related to the diffusivity and its dependence on Li concentration, the characteristic concentration dependent diffusion significantly contributes to the lithiation/delithiation hy steresis. These findings provide important insight into understanding the dynamics upon lithiation in Si anodes an d suggest methods to enhance the lithiation rate, which knowledge can be utilized to design batteries with enhanced rate performance. The irreversible changes that are inherent at atomic - scale upo n delithiation with rate effects was studied by utilizing the self - developed ReaxFF - based MD delithiation algorithm with controlled delithiation rate, which represent non - equilibrium (fast) and near - equilibrium (slow) process. Fast delithiation led to the formation of dense Si network near the surface and nanoporosity inside the a - Li x Si, resulting in 141 % volume dilation and significant amount of Li trapped inside (a - Li 1.2 Si) at the end of delithiation process. In contrast, slow delithiation proceeds 107 with no permanent inner pore with nearly Li - free structure (a - Li 0.2 Si) and minimal volume dilation (44 %). Regardless of the delithiation rate and existence of the trapped Li, continuous loss of the directly bonded Si - Si pairs caused the volume of the delithiat ed a - Li x Si to be higher than the equilibrium structure with the same Li concentration, resulting in faster lithiation rate in the subsequent cycle. Based on the atomistic modeling and the quantified degradation mechanism, battery operating guidelines, incl uding the delithiation rate and the depth of charge to avoid trapped Li and coating delamination, were suggested to improve the durability Si electrodes. The application of the reactive force field used in our lithiation and delithiation simulation (Li/Si and Li/Al/Si/O/H system) to SiO x system was evaluated by calculation and comparison of thermodynamic stability of SiO structures with DFT calculations. The formation energy of SiO respect to Si and SiO 2 was calculated to be a positive value for DFT calcula tions, whereas determined to be a negative value for ReaxFF calculations. Discrepancies between the thermodynamics stabilities originated from the training set of the ReaxFF, which only included various Si and SiO 2 phases. Since ReaxFF can only capture the configuration space described in the training set, I propose a guideline to extend of training set with metastable a - SiO x phases to cover the composition range of SiO x with x = 0 ~ 2. Based on the limitation on the transferability of the reactive force fi eld, I designed and construct an accurate and extensive training set for Li - Si system well - suited for development of machine - learning potentials. First principle DFT calculation (Ab Initio MD quenching and optimization) were performed to generate training set with c - Li x Si and a - Li x Si with x = 0 ~ 3.75. The liquid - like a - Li x Si structures at high temperature to ground - state a - Li x Si were all included in the training set, covering the entire configurational space for metastable a - Li x Si. The systematic strategy to construct an extensive training set can be further utilized to develop machine - learning 108 potential for different systems, which quality emulating the accuracy of the first - principle calculations and efficiency of the f orce f ields. 109 APPENDIX 110 APPENDIX Delithiatio n Algorithm 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 Firework Codes (Generation of Automated Training Set) Global Setting 126 Generate Firework Jobs 127 128 Generate L AMMPS Job 129 130 Extract Structures from L AMMPS Trajectory 131 132 133 Generate V ASP Jobs 134 Extract Energies from VASP 135 Collect Structures and Energies into Database 136 137 REFERENCES 138 REFERENCES (1) Tarascon, J. M.; Armand, M. Issues and Challenges Facing Rechargeable Lithium Batteries. 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